Applied Functional AnalysisLecture NotesSpring, 2010

Dr. H. T. BanksCenter for Research in Scientific Computation

Department of MathematicsN. C. State University

April 12, 2010

8 Example 6: Cantilever Beam

We introduce another example which is ubiquitous in structural applicationsand is ideal for use in illustrating our weak formulations that follow.

8.1 The Beam Equation

We consider the beam equation given by

�∂2y

∂t2+

∂y

∂t+

∂2

∂�2M = f(t, �) (1)

where the term ∂y∂t represents the external damping (the air or viscousdamping), M is the internal moment, � is the mass density, and f is theexternal force applied. For a development of this equation from basic prin-ciples, see [BSW, BT]. Before discussing the equation further, we discussboundary and initial conditions.Boundary Conditions We consider a beam with one end fixed and one endfree.

Fixed end:y(t, 0) = 0 zero displacement,∂y∂� (t, 0) = 0 zero slope.

Free end:M(t, l) = 0 zero moment,∂M∂� (t, l) = 0 zero shear.

We observe that the shear force is represented by S(t, �) = −∂M∂� (t, �).

Initial Conditions We consider a beam with initial displacement and velocitygiven by Φ and Ψ, respectively.

y(0, �) = Φ(�)y(0, �) = Ψ(�)

We observe that we have an equation (1) given with two unknowns;however we couple these with a constitutive relationship given by

M = M(y).

Using Hooke’s law (� = E� where � is the stress and � is the strain, E is theYoung’s modulus), we obtain (using basic principles [BT]) for linear elasticsystems the relationship

M = EI∂2y

∂�2

1

where I represents the moment of inertia of the cross-sectional area. Weremark that either E and/or I may be a function of �, i.e., E = E(�)and/or I = I(�).

We remark that the above derivations use the assumptions of linearelasticity and small displacements. Without small displacements, M couldbe a nonlinear function of y. For further discussions, see [BSW, BT].

We also have internal damping involved for which we must account. Wewill assume Kelvin-Voigt damping so that M(t, �) is given by

M(t, �) = EI∂2y

∂�2+ cDI

∂3y

∂�2∂t. (2)

Combining equations (1) and (2), we have the beam equation given by

�∂2y

∂t2+

∂y

∂t+

∂2

∂�2

(EI

∂2y

∂�2+ cDI

∂3y

∂�2∂t

)= f(t, �),

with the modified boundary conditions(EI ∂

2y∂�2

+ cDI∂3y∂�2∂t

)∣�=l = 0,

[∂∂�

(EI ∂

2y∂�2

+ cDI∂3y∂�2∂t

)]�=l

= 0.

8.2 The Beam Equation in the form x = Ax+ F

We wish to write the beam equation in the form x = Ax + F . Let x berepresented by

x(t, ⋅) =

(y(t, ⋅)y(t, ⋅)

)and take as state space X = H2

L(0, l)× L2(0, l) where

H2L(0, l) = {' ∈ H2(0, l)∣ '(0) = 0, '′(0) = 0}.

Thus for elements in H2L(0, l), the function and the derivative both vanish

at the left boundary. We rewrite the equation using the abbreviation ∂ =∂∂� , ⋅ =

∂∂t , and have

y =1

�

[−∂2(EI∂2y)− ∂2(cDI∂

2y)]−

�y +

1

�f.

2

We rewrite this in first order vector form as

d

dt

(yy

)=

(0 I−A −B

)(yy

)+

(0

1�f

).

Let

A =

(0 I−A −B

)where we define A and B in the following way:

A� = 1�∂

2(EI∂2�)

B� = 1�∂

2(cDI∂2�) +

��.

Then we have the form x = Ax+ F with

D(A) = {(', ) ∈ X∣ ∈ H2L(0, l), A'+B ∈ L2(0, l),

[EI∂2'+ cDI∂2 ]l = 0, [∂(EI∂2'+ cDI∂

2 )]l = 0}.

8.2.1 A is an Infinitesimal Generator

We claim that A is an infinitesimal generator of a C0 semigroup in Xℰ whereXℰ is an associated energy space. Specifically, we define Xℰ as X with theenergy product

⟨('1, 1), ('2, 2)⟩ℰ =∫ l

0 EI∂2'1∂

2'2d� +∫ l

0 � 1 2d�

=∫ l

0 EI∂2'1∂

2'2d� + ⟨� 1, 2⟩.

If 0 < �1 ≤ �(�) ≤ �2 < ∞, then ⟨� 1, 2⟩ is equivalent to the norm inL2(0, l) where the norm for H2(0, l) is defined by

∣'∣2H2 = ∣'∣2L2 + ∣'′∣2L2 + ∣'′′∣2L2 .

An equivalent norm is

∣'∣2∼ = ∣'(a)∣+ ∣'′(b)∣+ ∣'′′∣2L2 ,

where a,b are any values in [0, l]. In particular, on H2L, we can take ∣'∣2H2 ≡

∣'(0)∣+ ∣'′(0)∣+ ∣'′′∣2L2 . This defines a Hilbert space which is topologicallyequivalent to X so that we thus have that if A generates a C0 semigroup inXℰ , it is also a C0 semigroup in X.

3

8.2.2 Poincare (First) Inequality

To argue that A is a generator, we need a standard and well-known [A]inequality given by

Suppose there exists a set Ω which is a bounded subset of Rn. Then thereexists a c = c(Ω), such that for all ' ∈W l,2

0 (Ω) = H l0(Ω), one has

∣'∣2Hl(Ω) ≤ c(Ω)∑∣s∣=l

∫Ω∣∂s'∣2

where s = (s1, . . . , sn).

8.2.3 Dissipativeness of A

We argue that A is dissipative in Xℰ . Recalling that M = EI∂2'+cDI∂2 ,

we have

⟨A(', ), (', )⟩ℰ =

∫ l

0EI∂2 ∂2'+

∫ l

0

[−∂2(EI∂2'+ cDI∂

2 )− ]

=

∫ l

0EI∂2 ∂2'−

∫ l

0∂2(EI∂2'+ cDI∂

2 ) −∫ l

0 2)

=

∫ l

0EI∂2 ∂2'−

∫ l

0

((EI∂2'+ cDI∂

2 )∂2 + 2)

− ∂M ∣l0 +M∂ ∣l0

= −∫ l

0cDI∣∂2 ∣2 −

∫ l

0 ∣ ∣2

≤ −k∣ ∣2H2(0,l)

≤ 0

for k =min {∣cDI∣, ∣ ∣}. Here we have integrated by parts twice and used thefact that (', ) ∈ D(A) so that (0) = ∂ (0) = 0 and M(t, l) = ∂M(t, l) =0.

8.2.4 ℛ(�I −A) = X for some �

To verify the range statement of Lumer Phillips, we need to show for some� that

�(', )−A(', ) = (ℎ, g) (3)

4

can be solved for (', ) for a given (ℎ, g) ∈ X. However, (3) reduces tofinding, for any (ℎ, g) ∈ X = H2

L(0, l) × L2(0, l), a solution (', ) in D(A)that satisfies

− + �' = ℎ (4)

A'+B + � = g (5)

where ℎ ∈ H2L and g ∈ L2. We can rewrite (4) to obtain

= �'− ℎ (6)

By substituting (6) into (5), we can reduce the above question to one ofsolving

�2'+A'+ �B' = g + �ℎ+Bℎ (7)

for ', given any (ℎ, g) in H2L(0, l)×L2(0, l). After solving for ', we can then

obtain using (6). If we obtain (', ) ∈ D(A), then we have proven therange statement of Lumer-Phillips. To complete the argument we will usethe concepts and results for sesquilinear forms to be discussed below.

5

9 Gelfand Triple and Sesquilinear Forms

An easy way to solve the above problem is to use the Lax-Milgram theorem;however, we first introduce Gelfand triples. For relevant material, see also[Sh, T, W].

9.1 Concept of Gelfand Triple

The usual notation for a Gelfand triple is “V ↪→ H ↪→ V ∗ with pivot spaceH”. This notation stands for V,H, complex Hilbert spaces, such that V ⊂ Hand V is densely and continuously embedded in H. That is, V is a densesubset of H and

∣v∣H ≤ k∣v∣Vfor all v ∈ V and some constant k. Therefore, you can identify elements inV with elements in H with an injection operator, i, where i is continuousand i(V ) is a dense subset of H.

We denote by V ∗ the conjugate dual of V . That is, V ∗ consists of allconjugate linear continuous functionals on V . (Note that we use V ′ to denotethe algebraic dual of V and V ∗ to denote the topological dual. Frequentlyone encounters exactly the opposite notation in the literature.)

For ℎ ∈ H, we define '(ℎ) ∈ V ∗ by

'(ℎ)(v) = ⟨ℎ, v⟩Hfor v ∈ V . We claim that ' : H → '(H) ⊂ V ∗ is continuous, linear,one-to-one and onto. Moreover, '(H) is dense in V ∗ in the V ∗ topology.

By the Riesz theorem, every ℎ∗ ∈ H∗ can be represented by

ℎ∗(ℎ) = ⟨ℎ∗, ℎ⟩H ℎ ∈ H

for some ℎ∗ ∈ H. Hence, it is readily argued that H∗ is isomorphic to H,H∗ ≃ H. That is, we may identify H∗ with H through '(H) = H ≃ H∗

and write ℎ(v) = ⟨ℎ, v⟩H for ℎ ∈ H = H∗.This construction is commonly written as V ↪→ H ∼= H∗ ↪→ V ∗ for the

pivot space H.

9.2 Duality Pairing

With a Gelfand triple, one frequently utilizes the duality pairing denoted by⟨⋅, ⋅⟩V ∗,V given by the extension by continuity of the H inner product fromH × V to V ∗ × V . That is, for v∗ ∈ V ∗,

v∗(v) = ⟨v∗, v⟩V ∗,V = limn⟨ℎn, v⟩H

6

where ℎn ∈ H,ℎn → v∗ in V ∗. Note that we have ⟨ℎ, v⟩V ∗,V = ⟨ℎ, v⟩H ifℎ ∈ V ∗ also satisfies ℎ ∈ H.

9.3 Sesquilinear Forms

Definition 1 Let H1 and H2 be two complex Hilbert spaces, and let � :H1 ×H2 → C. Then we call � a sesquilinear form if it satisfies

1. � is linear/conjugate linear;

2. � is continuous. More precisely, there exists > 0 such that

∣�(x, y)∣ ≤ ∣x∣1∣y∣2

for x ∈ H1 and y ∈ H2.

9.4 Norm of a Sesquilinear Form

The norm of a sesquilinear form � is defined by

∣�∣ = supx,y ∕=0

∣�(x, y)∣∣x∣1∣y∣2

for x ∈ H1 and y ∈ H2.

9.5 Representation

We consider two different representations:

1. For y ∈ H2, x → �(x, y) is continuous and linear on H1. Therefore,by the Riesz Theorem, there exists a unique z ∈ H1 such that

�(x, y) = ⟨x, z⟩H1 .

Thus there exists a mapping B ∈ ℒ(H2, H1) defined by By = z. There-fore,

�(x, y) = ⟨x, z⟩H1 = ⟨x,By⟩H1 .

We find that ∣�∣ = ∣B∣ℒ(H2,H1). To see this we argue as follows.

7

For x in H1 and y in H2, we have

∣�∣ = supx,y ∕=0

∣�(x,y)∣∣x∣1∣y∣2 = sup

∣⟨x,By⟩H1∣

∣x∣1∣y∣2

≤ sup ∣x∣1∣By∣1∣x∣1∣y∣2 = supy ∕=0

∣By∣1∣y∣2

= ∣B∣ℒ(H1,H2).

Therefore, we have ∣�∣ ≤ ∣B∣ℒ(H2,H1). Thus we need only argue that∣�∣ ≥ ∣B∣ℒ(H2,H1). However, we have

∣�∣ = supx,y ∕=0

∣�(x,y)∣∣x∣1∣y∣2 = sup

∣⟨x,By⟩H1∣

∣x∣1∣y∣2

≥ supx=By

∣⟨By,By⟩H ∣∣By∣1∣y∣2 = sup ∣By∣1∣y∣2

= ∣B∣ℒ(H2,H1).

Hence we have ∣�∣ = ∣B∣ℒ(H2,H1).

2. For x ∈ H1, y → �(x, y) is continuous and antilinear on H2, i.e.,�(x, ⋅) ∈ H∗2 . Therefore, by the Riesz Theorem, there exists a uniquez ∈ H2 such that

�(x, y) = ⟨z, y⟩H2 .

Thus there exists a mapping A ∈ ℒ(H1, H2) defined by Ax = z. There-fore,

�(x, y) = ⟨z, y⟩H2 = ⟨Ax, y⟩H2 ,

with ∣A∣ = ∣�∣. Define �(y, x) = �(x, y), and interchange the roles ofH1 and H2 in the above arguments to argue ∣A∣ = ∣�∣.

We wish use our Gelfand triple and sesquilinear form formulations toconsider elliptic equations of the form Ax = f . However, first, we begin bydiscussing operators of the form A : H1 → H2 such that A is bounded andcontinuous. This is very useful in integral equations. However, we will needto modify this formulation to account for the unbounded nature of PDE’s.We can think of our equation Ax = f in the form �(x, y) = ⟨f, y⟩H for all yin H where � is equivalent to A. In other words, ⟨Ax− f, y⟩H = 0 for all yin H. The useful results are given in the famous Lax-Milgram theorems, inbounded and unbounded formulations.

8

10 Lax-Milgram(bounded form)

Theorem 1 (Lax-Milgram Theorem (bounded form)) Suppose � : H×H →C is continuous and linear/conjugate linear, i.e., it is a sesquilinear formwith

∣�(x, y)∣ ≤ ∣x∣H ∣y∣Hfor all x and y in H, and � is strictly positive, i.e.,

∣�(x, x)∣ ≥ �∣x∣2

for all x in H where and � are positive constants. Then there existsA : H → H,A ∈ ℒ(H,H) defined by

�(x, y) = ⟨Ax, y⟩

with ∣A∣ = ∣�∣ ≤ for all x and y in H. Moreover,

�(A−1x, y) = ⟨x, y⟩

with ∣A−1∣ ≤ 1� for all x and y in H.

Proof of Theorem

By the Riesz Theorem, we know there exists A ∈ ℒ(H) such that ∣A∣ =∣�∣ ≤ where �(x, y) = ⟨Ax, y⟩H for all x and y in H.

We claim that A is one-to-one. For this we need to show that Ax = 0implies x = 0. Suppose Ax = 0. Then, by the definition of A and theassumptions given, we have

0 = ∣⟨Ax, x⟩∣ = ∣�(x, x)∣≥ �∣x∣2.

Therefore, �∣x∣2 ≤ 0 implies ∣x∣ = 0, and hence x = 0. In other words, A isone-to-one. Since A is one-to-one, we know that A−1 exists on ℛ(A) ⊂ H.

We next claim that A−1 is bounded on ℛ(A). By the proposition that� was strictly positive and the definition of A, we have

�∣x∣2 ≤ ∣�(x, x)∣ = ∣⟨Ax, x⟩∣≤ ∣Ax∣∣x∣.

Therefore, ∣Ax∣ ≥ �∣x∣ for all x in H. In particular, for x = A−1y, we have

∣y∣ ≥ �∣A−1y∣.

9

Therefore,

∣A−1y∣ ≤ 1

�∣y∣

for all y in ℛ(A). In other words, A−1 is bounded. Finally all we have toshow is that ℛ(A) = H. By assumption, we know that A is a continuousoperator and we have argued that A−1 is bounded on ℛ(A). Therefore,ℛ(A) is closed. Now suppose ℛ(A) ∕= H. This implies there exists z ∕= 0such that z ⊥ ℛ(A). In other words,

⟨Ax, z⟩ = 0

for all x in H. In particular, choosing x = z, we have

0 = ⟨Az, z⟩ = �(z, z) ≥ �∣z∣2.

This implies z = 0; therefore, we have a contradiction. So, ℛ(A) = H.

10.1 Discussion of Ax = f with A bounded

The bounded form of Lax-Milgram is very useful in linear integral equations.The Fredholm integral equations with kernel k ∈ L2([a, b]× [a, b]) are givenby ∫ b

ak(x, y)'(y)dy = f(x) x ∈ [a, b] (first kind)

'(x)−∫ b

ak(x, y)'(y)dy = f(x) x ∈ [a, b] (second kind).

These can be written as operator equations in H = L2(a, b),

A' = f

'−A' = f

to be solved for ', given f , where A is a bounded linear operator (discussedbelow) in H. It also has applications in scattering theory (acoustic andelectromagnetic radiation), far field radiation, and single and double layerpotential theory.

For relevant material, see also [K], [CK1], and [CK2].As we have noted, the bounded form of Lax-Milgram is adequate if one

wants to solve Ax = f in H where A is bounded. Consider k ∈ L2(Ω) withΩ = [0, 1]× [0, 1] with H = L2(0, 1). Then define A : H → H by

(A')(t) =

∫ 1

0k(t, �)'(�)d�.

10

We can see that this operator is bounded, i.e,∫ 1

0

(∫ 1

0k(t, �)'(�)d�

)2

dt <

∫ 1

0∣'(�)∣2d�.

Therefore, there is a sesquilinear form � such that A corresponds � as inthe Lax-Milgram above with

�(', ) =∫ 1

0

(∫ 10 k(t, �)'(�)d�

) (t)dt

= ⟨A', ⟩L2(0,1).

We can show that all the requirements of Lax-Milgram (bounded form)are met with this operator A; therefore, Lax-Milgram (bounded form) isapplicable.

10.2 Example - The steady state heat equation

Let Ω = [0, 1]× [0, 1]. Then the heat equation is given by

∂u

∂t= ∇ ⋅ (D∇u) + f

where ∇ = ∂∂�1i + ∂

∂�2j with (�1, �2) ∈ Ω. We assume boundary conditions

that require u ∈ H10 (Ω) Then the steady state equation is given by

−∇ ⋅ (D∇u) = f (8)

or∂

∂�1(D

∂u

∂�1) +

∂

∂�2(D

∂u

∂�2) = f(�1, �2).

In the weak or variational form, we have (by integration by parts and useof the boundary conditions)

⟨D∇u,∇'⟩L2 = ⟨f, '⟩L2 (9)

for a test function ' ∈ H10 (Ω). We define � on H1

0 × H10 by �( ,') =

⟨D∇ ,∇'⟩L2 with ∣D∣ ≤ , i.e., D is bounded. Then � is not continuouson H = L2(Ω). On the other hand, for ', ∈ H1

0

∣�( ,')∣ ≤ ∣∇ ∣L2 ∣∇'∣L2

≤ ∣ ∣H10∣'∣H1

0

11

implies � is continuous on V = H10 (Ω).

Moreover, we do have some type of positivity. If ∣D(�1, �2)∣ ≥ �, then

∣�(',')∣ = ∣⟨D∇',∇'⟩∣

≥ �∣∇'∣2L2≥ �∣'∣H1

0 (Ω).

In other words, in the V = H10 (Ω) norm, we would have both continuity and

strict positivity of the sesquilinear form. But we don’t have continuity andstrict positivity in the H = L2(Ω) sense. Thus if we choose our H space tobe H1

0 (Ω), we would be guaranteed a unique solution of ⟨Au− f, '⟩H10

= 0.

That is, u = A−1f in the H10 sense (and of course u ∈ H1

0 since A−1 isbounded from H1

0 to H10 . This also requires the input function f to be

in H10 (Ω) which is a strong requirement for f . Therefore, it would not be

useful in some cases to choose our H space to be H10 (Ω) in the Lax-Milgram

theorem. Instead, we need an extension of Lax-Milgram to treat the heatequation in a more reasonable manner.

12

11 Lax-Milgram (unbounded form)

Let V ↪→ H ↪→ V ∗ be a Gelfand triple.

Definition 2 A sesquilinear form � is said to be V -continuous if

∣�(', )∣ ≤ ∣'∣V ∣ ∣V

for all ' and in V .

Consequences of V -continuity

For a fixed ' in V , we consider the mapping → �(', ) for a V -continuous�. This mapping is continuous by the definition of continuity above, andits a conjugate linear mapping into C. Therefore, → �(', ) is in V ∗.This implies there exists an operator A ∈ ℒ(V, V ∗) such that �(', ) =⟨A', ⟩V ∗,V .

Conversely, if A ∈ ℒ(V, V ∗), we can define � : V × V → C by �(', ) =(A') so that � is V -continuous and linear/conjugate linear.

In other words, if we have a V -continuous sesquilinear form �, there is aone-to-one correspondence between � and A ∈ ℒ(V, V ∗). While the operatoris not bounded in V , it will be useful in considering the equation Au = f inV ∗ will be useful. (Note the weak requirements on f in this case.)

Definition 3 A sesquilinear form � is V -coercive if there exists a constant� > 0 such that

∣�(',')∣ ≥ �∣'∣2Vfor ' ∈ V .

Theorem 2 (Lax-Milgram Theorem (unbounded form)) Let V ↪→ H ↪→ V ∗

be a Gelfand triple. Let � : V × V → C be a V -continuous, V -coercivesesquilinear form. Then A : V → V ∗ given by

�(', ) = ⟨A', ⟩V ∗,V

is a linear (topological) isomorphism between V and V ∗. Moreover, A−1 iscontinuous from V ∗ to V with

∣A−1∣ℒ(V ∗,V ) ≤1

�.

13

Proof of Theorem

Let R : V ∗ → V be a Riesz isomorphism, so that for v∗ ∈ V ∗ we have

v∗(v) = ⟨v∗, v⟩V ∗,V = ⟨Rv∗, v⟩V . (10)

Then R−1 : V → V ∗ is continuous.Let � : V × V → C be a V -continuous, V -coercive sesquilinear form.

Then → �(', ) in V ∗ implies there exists a z in V such that �(', ) =⟨z, ⟩V . From the bounded version of Lax-Milgram, this implies there existsA ∈ ℒ(V ) such that

�(', ) = ⟨A', ⟩V (11)

for all ' and in V , with A one-to-one, A onto, ∣A∣ℒ(V ) ≤ and ∣A−1∣ℒ(V ) ≤1� . In other words, A is an isomorphism V → V .

We have that R−1 : V → V ∗ is also an isomorphism; therefore, R−1A :V → V ∗ is an isomorphism. The claim is that A = R−1A. By (10) we havefor all ', ∈ V

⟨R−1A', ⟩V ∗,V = ⟨A', ⟩V .

However, by (11), this implies

⟨R−1A', ⟩V ∗,V = �(', ).

Therefore, by definition of A : V → V ∗, A must be given by A = R−1A.So, we have

⟨A', ⟩V ∗,V = �(', ) ≤ ∣'∣V ∣ ∣V (12)

and⟨A','⟩V ∗,V ≥ �∣'∣2V . (13)

Recall that sup∣ ∣=1 ∣⟨A', ⟩V ∗,V ∣ = ∣A'∣V ∗ . Letting ' = A−1 in (13) andcombining this with (12) and (13), we have

�∣'∣V ≤ ∣A'∣V ∗ ≤ ∣'∣V . (14)

Therefore, we have∣A∣ℒ(V,V ∗) ≤

and

∣A−1∣ℒ(V ∗,V ) ≤1

�.

14

Implications of Lax-Milgram(unbounded form)

For A as in the Lax-Milgram Theorem 2 and f ∈ V ∗, we consider Au = fin V ∗. Then Lax-Milgram implies there exists a unique solution u = A−1fin V that depends continuously on f . More precisely,

∣u∣V = ∣A−1f ∣V ≤ �∣f ∣V ∗ .

We revisit the steady-state heat equation (8) in the example above with D ∈L∞(Ω) and V = H1

0 (Ω). This, of course, allows discontinuous coefficients.Then for any f in V ∗ = H−1(Ω) there exists a unique u ∈ V satisfyingAu = f . That is, ⟨Au− f, '⟩V ∗,V = �(u, ')− ⟨f, '⟩V ∗,V = ⟨D∇u,∇'⟩L2 −⟨f, '⟩V ∗,V = 0 for all ' ∈ V . Thus we say u ∈ V satisfies Au = f in thesense of V ∗. Hence (8) holds in the V ∗ sense which is precisely (9). This isalso sometimes referred to as u satisfying (8) in the “sense of distributions.”

11.1 The concept of DA

We assume that we have a Gelfand triple V ↪→ H ↪→ V ∗ and a continuoussesquilinear form � : V × V → C. As usual, f( ) = ⟨f, ⟩H defines, forf ∈ H, an element f ∈ V ∗. Considering (A')( ) = �(', ) for ' and inV , we define

DA = {' ∈ V ∣A' ∈ H}.That is, DA is the set of ' ∈ V such that A' ∈ V ∗ has the representation(A')( ) = ⟨', ⟩H , for in V and for some ' in H.

We denote this element ' by −A' = ', i.e., A is linear from DA ⊂ Vinto H and given by

�(', ) = ⟨−A', ⟩Hfor in V and ' in DA.

We note that the above can be interpreted as: ' ∈ DA ⊂ V if andonly if ' ∈ V and A' ∈ H∗ ∼= H so that (A')( ) = ⟨', ⟩H , for all in V , and for some ' ∈ H. Alternatively, we may write DA = {' ∈V ∣∣(A')( )∣ = ∣�(', )∣ ≤ k∣ ∣H , ∈ V } for some k. Moreover, we couldwrite DA = {' ∈ V ∣ → �(', ) is in H∗, i.e, continuous on H (continuousin the H sense) }.

Note that we also have

�(', ) = (A')( ) = ⟨A', ⟩V ∗,Vfor all ' and in V . If we restrict ' ∈ DA then this also equals ⟨−A', ⟩H .

Theorem 3 If � is a V -continuous V -coercive sesquilinear form on V , thenDA is dense in V and, hence, dense in H.

15

Proof of Theorem

Define �(', ) = �( ,') (called the “adjoint” sesquilinear form)If � is V -coercive and V continuous, then � is also V -coercive and V con-tinuous. In other words, there exists an operator A : V → V ∗ such thatA ∈ ℒ(V, V ∗) with

�(', ) = ⟨A', ⟩V ∗,V = (A')( )

for all ' and in V . Hence, ℛ(A) = V ∗ by the Lax-Milgram theorem.

Next we need to show that DA is dense in V . It suffices to show that iff ∈ V ∗ and f(v) = 0 for all v ∈ DA, then f ≡ 0.Let f ∈ V ∗ such that f(v) = 0 for all v ∈ DA. Since f ∈ V ∗ andℛ(A) = V ∗,then there exists ' ∈ V such that f = A'.For v ∈ DA, we have

(−Av)(') = (Av)(') = �(v, ')

= �(', v) = ⟨A', v⟩V ∗,V

= ⟨f, v⟩V ∗,V = f(v)

= 0.

Therefore, (Av)(') = 0 for all v ∈ DA.But, we know ℛ(A∣DA) = H ∼= H∗. Then, for every ℎ ∈ H∗, ℎ(') = 0 .However, as H∗ is dense in V ∗, then this implies ' = 0. Moreover, A' = fimplies f = 0.

Thus we find that any V -continuous V -coercive sesquilinear form on Vgives rise to a densely defined operator A on D(A) = DA with

�(', ) = ⟨A', ⟩V ∗,V ', ∈ V= ⟨−A', ⟩H ' ∈ DA, ∈ V.

11.2 V-elliptic

Definition 4 A sesquilinear form � on V is V -elliptic if there exists a con-stant � > 0 such that

Re �(',') ≥ �∣'∣2V ' ∈ V.

16

Discussion of V -elliptic

We note that � is V -elliptic implies � is V -coercive. Of course, if we areworking in real spaces, Re � = � and V -elliptic is equivalent to V -coercive.We also remark that the terminology among various authors is not standard.Some authors (e.g., Wloka in [W]) use our definition for V -coercive as thedefinition of V -elliptic and then use

∣�(',') + k∣'∣2H ∣ ≥ �∣'∣2V ' ∈ V, � > 0

as the definition of V -coercive.Some of the terminology and usage of sesquilinear forms derives directly

from that for PDE’s of the form

∂y

∂t=∑i,j

∂

∂�i(aij

∂y

∂�j) +

∑j

bj∂y

∂�j, t > 0, � ∈ G ⊂ Rn.

Definition 5 In classical PDE’s, an “operator” {aij} is said to be stronglyelliptic on G if there exists � > 0 such that for � ∈ G

Re∑i,j

aij(�)qiqj ≥ �∑i

∣qi∣2

for all q ∈ Cn.

An associated sesquilinear form on V = H1(G) can be defined by

�(', ) =

∫G

⎡⎣∑i,j

aij(�)∂'

∂�i

∂

∂�j+∑k

bk∂'

∂�k

⎤⎦ d�Discussion of Strongly elliptic

It is a standard result that {aij} strongly elliptic implies there exists � > 0such that for � sufficiently large

Re �(',') + �∣'∣2H ≥ �∣'∣2V , ' ∈ V.

Hence if {aij} is strongly elliptic, then for some �0 sufficiently large,

�(', ) = �(', ) + �0⟨', ⟩H

is V -elliptic and hence V -coercive.A most useful result is that continuous V -elliptic forms give rise to oper-

ators that are infinitesimal generators of C0 (actually analytic) semigroups.

17

12 Some Useful Theorems

Definition 6 A semigroup T (t) on H is called an analytic semigroup ift→ T (t)' is analytic for each ' in H.

Theorem 4 Let V , H be complex Hilbert spaces with V ↪→ H ↪→ V ∗ andsuppose that � : V × V → C is continuous and V -elliptic; i.e.

∣�(', )∣ ≤ ∣'∣V ∣ ∣V ', ∈ V,

andRe �(',') ≥ �∣'∣2V � > 0, ' ∈ V.

Define A : D(A) ⊂ V → H by

D(A) = {' ∈ V ∣ there exists K' > 0 such that �(', ) ≤ K'∣ ∣H , ∈ V }

and�(', ) = ⟨−A', ⟩H , ' ∈ D(A), ∈ V.

Then D(A) is dense in H and A is the infinitesimal generator of a contrac-tion semigroup in H that actually is an analytic semigroup.

The proof of this theorem will come later.

Theorem 5 Suppose all the assumptions of Theorem 4 hold except that theV -ellipticity condition for � is replaced by

Re �(',') + �0∣'∣2H ≥ �∣'∣2V

for some �0, � > 0, ' ∈ V . Then defining A as in Theorem 4, we havethat A is densely defined and is the infinitesimal generator of an analyticsemigroup in H.

12.1 Return to Example 1

The system is given by

∂y

∂t=

∂

∂�

(D(�)

∂y

∂�

)y(t, 0) = 0, ∂y

∂� (t, l) = 0

y(0, �) = Φ(�).

18

We choose the state space X = L2(0, l) as before. To obtain the weakvariational form and the space V , we work backwards by multiplying theequation by a ”test” function ' and integrating.∫ l

0 y' =∫ l

0(Dy′)′'

=∫ l

0 −Dy′'′ +Dy′'∣l0

Therefore we have

⟨y(t), '⟩+ ⟨Dy′(t), '′⟩ −Dy′(t)'∣l0 = 0 (15)

where ⟨⋅, ⋅⟩ is the usual inner product in L2. However, (15) is equivalent to

⟨y(t), '⟩+ ⟨Dy′(t), '′⟩ = 0

if ' ∈ H1L(0, l) = {' ∈ H1(0, l)∣'(0) = 0} and Dy′(t, l) = 0.

Defining V = H1L(0, l) and � on V × V by

�(', ) = ⟨D'′, ′⟩,

we may write the equation in weak form as: find y(t) ∈ V satisfying

⟨y(t), '⟩+ �(y(t), ') = 0

for all ' ∈ V . This equation is equivalent to the original system whenevery(t) ∈ V

∩H2(0, l) by using the reverse of the above arguments.

Next we consider the flux boundary condition of the original problem.Suppose y(t) is a weak solution, i.e.,

⟨y(t), '⟩+ �(y(t), ') = 0 ∀ ' ∈ V

y(0) = Φ(�)

and y in H2(0, l). Then

⟨y(t), '⟩+

∫ l

0Dy′'′ = 0.

Integrating by parts, we find that the above equation is equivalent to∫ l

0(y − (Dy′)′)'+Dy′(t, l)'(l) = 0 (16)

19

for all ' ∈ H1L. However, H1

0 ⊂ H1L; therefore,∫ l

0(y − (Dy′)′)' = 0 (17)

for all ' ∈ H10 . Since H1

0 is dense in L2(0, l), this implies y − (Dy′)′ =0. However, if we choose ' ∈ H1

L such that '(l) ∕= 0, then (16) impliesDy′(t, l) = 0, i.e., the flux boundary condition is satisfied.

If we define the V -inner product as

⟨', ⟩V =

∫ l

0'′ ′,

and set H = X = L2(0, l), then we readily see V ↪→ H ↪→ V ∗. Note thatthe V norm is equivalent to the usual H1 norm on H1

L(0, l). Furthermore,we have

∣�(', )∣ = ∣⟨D'′, ′⟩∣

≤ ∣D∣∞∣'′∣L2 ∣ ′∣L2

= ∣D∣∞∣'∣V ∣ ∣V .

Also,Re �(',') = Re ⟨D'′, '′⟩ ≥ �∣'′∣2L2 = �∣'∣2V

so that � is bounded and V -elliptic.We can define A : V → V ∗ by

⟨A', ⟩V ∗,V = �(', ) = ⟨D'′, ′⟩.

Note that A' ∈ H ↪→ V ∗ if and only if ⟨D'′, ′⟩ = ⟨w, ⟩ for all ∈ V forsome w ∈ H. However, integrating by parts we have∫ l

0 D'′ ′ = −

∫ l0(D'′)′ +D'′ ∣l0

= ⟨−(D'′)′, ⟩+D(l)'′(l) (l)

= ⟨−(D'′)′, ⟩

if '′(l) = 0 and (D'′)′ ∈ L2(0, l). Thus we may define

A' = (D'′)′

20

onD(A) = {' ∈ H1

L(0, l)∣(D'′)′ ∈ L2(0, l), '′(l) = 0}

and obtain A' = −A' ∈ H exactly whenever ' ∈ D(A).The above results hence guarantee that A generates a C0-semigroup

(actually an analytic semigroup) T (t) on H = X = L2(0, l).

12.2 Return to Example 2

We consider again the transport equation given by

∂y∂t + ∂

∂� (�y) = ∂∂� (D ∂y

∂� )− �yy(t, 0) = 0

(D ∂y∂� − �y)∣�=l = 0

y(0, �) = Φ(�).

We can rewrite the transport equation by

yt = (Dy′ − �y)′ − �y.

Multiplying by a test function and integrating from 0 to l, we have

⟨yt, '⟩ =∫ l

0 ((Dy′ − �y)′'− �y') d�

= −⟨Dy′ − �y, '′⟩+ (Dy′ − �y)'∣l0 − ⟨�y, '⟩.

If we choose H = X = L2(0, l) and V = H1L(0, l) as in Example 1 above,

with the same V inner product, we have

⟨yt, '⟩ = −⟨Dy′ − �y, '′⟩ − ⟨�y, '⟩.

As before, we have V ↪→ H ↪→ V ∗. Then we can define the sesquilinear form� : V × V → C by

�(', ) = ⟨D'′ − �', ′⟩+ ⟨�', ⟩.

Therefore, we have the equation

⟨y, '⟩+ �(y, ') = 0.

Briefly, we discuss the various possibilities for boundary conditions andthe effects on the choice of V . If we had a no flux boundary condition at� = 0, we would choose V = H1

R(0, l). On the other hand, if we had essential

21

boundary conditions at both boundaries, i.e., y = 0 at � = 0 and � = l, wewould need to choose V = H1

0 (0, l). A third possibility is if we had the noflux boundary conditions at both boundaries, � = 0 and � = l. In that case,as both boundary conditions were natural, we would choose V = H1(0, l).

The V -continuity of � is established by arguing

∣�(', )∣ ≤ ∣D∣∞∣'′∣H ∣ ′∣H + ∣�∣∞∣'∣H ∣ ′∣H + ∣�∣∞∣'∣H ∣ ∣H

≤ ∣D∣∞∣'∣V ∣ ∣V + ∣�∣∞k∣'∣V ∣ ∣V + ∣�∣∞k2∣'∣V ∣ ∣V

= (∣D∣∞ + k∣�∣∞ + k2∣�∣∞)∣'∣V ∣ ∣V .

As � is V -continuous, we have

�(', ) = ⟨A', ⟩V ∗,V ' ∈ V= ⟨−A', ⟩H ' ∈ D(A),

where D(A) is defined by

D(A) = {' ∈ H2(0, l)∣'(0) = 0, (D'′ − �') ∈ H1(0, l), (D'′ − �')(l) = 0}.

Note that V carries the essential boundary conditions, while the naturalboundary conditions are found in D(A).

To show that � is V -coercive, if we assume D ≥ c1 > 0 and ⟨�', '⟩ ≥−∣�∣∞∣'∣2H , then we have

Re �(',') ≥ c1∣'∣2V −∣�∣2∞

4� ∣'∣2H − �∣'∣2V − ∣�∣∞∣'∣2H

= (c1 − �)∣'∣2V − ( ∣�∣2∞

4� + ∣�∣∞)∣'∣2H .

Hence, setting � = c12 , we have

Re �(',') ≥ c1

2∣'∣2V − �0∣'∣2H

for �0 = ∣�∣2∞2c1

+ ∣�∣∞. Thus we see that � given by

�(', ) = �(', ) + �0⟨', ⟩= ⟨−A', ⟩+ �0⟨', ⟩= ⟨−(A− �0)', ⟩

is V -elliptic (indeed it is V coercive). We thus find that A− �0, and henceA, is the generator of an analytic semigroup in H = X = L2(0, l).

22

12.3 Return to Example 6

We return to the beam equation. Recall the system is given by

�ytt + yt + ∂2M = f 0 < � < l,

with

y(t, 0) = 0 =∂y

∂�(t, 0)

M(t, l) = 0 = ∂M(t, l),

where M(t, �) = EI∂2y + cDI∂2yt. We choose as our basic space H =

L2(0, l) with the weighted inner product ⟨', ⟩H = ⟨�', ⟩L2(0,l). Then theweak form becomes

⟨ytt +

�yt, '⟩H + ⟨EI

�∂2y, ∂2'⟩H + ⟨cDI

�∂2yt, ∂

2'⟩H = ⟨1�f, '⟩H

for all ' ∈ V = H2L(0, l). We choose the weighted inner product for V given

by ⟨', ⟩V =∫ l

0 EI'′′ ′′.

We define the sesquilinear forms �1 and �2 on V × V → C by

�1(', ) = ⟨EI�'′′, ′′⟩H =

∫ l

0EI'′′ ′′

�2(', ) = ⟨cDI

�'′′, ′′⟩H + ⟨

�', ⟩H .

The weak form of the equation is then

⟨ytt, '⟩H + �1(y, ') + �2(yt, ') = ⟨f�, '⟩H

for ' ∈ V . To write this in first order vector form, we use the state spaceXE = ℋ = V ×H with the space V = V × V , noting that V ↪→ H ↪→ V ∗

and V ↪→ ℋ ↪→ V∗ form Gelfand triples, where V∗ = V × V ∗.

Homework Exercises

∙ Ex. 9 : Explain why we have V∗ = V ×V ∗ in the Gelfand triple insteadof V∗ = V ∗ × V ∗.

23

We define the sesquilinear form � : V × V → C by (for � = (', ), � =(g, ℎ) in V)

�(�, �) = �((', ), (g, ℎ)) = −⟨ , g⟩V + �1(', ℎ) + �2( , ℎ).

Using the state variable w(t) = (y(t, ⋅), yt(t, ⋅)) in XE = ℋ, we can rewritethe equation as

⟨w(t), �⟩ℋ + �(w(t), �) = ⟨F (t), �⟩ℋ

for � ∈ V, where F (t) = (0, 1�f(t)).

We readily argue that � is bounded (continuous) and V-elliptic (actually,� − �0∣ ⋅ ∣2XE is V-elliptic). Consider first the boundedness argument:

∣�(�, �)∣ = ∣�((', ), (g, ℎ))∣ = ∣ − ⟨ , g⟩V + �1(', ℎ) + �2( , ℎ)∣

≤ ∣ ∣V ∣g∣V + 1∣'∣V ∣ℎ∣V + 2∣ ∣V ∣ℎ∣V≤ ∣�∣V ∣�∣V + 1∣�∣V ∣�∣V + 2∣�∣V ∣�∣V

= (1 + 1 + 2)∣�∣V ∣�∣V

for �, � ∈ V. The arguments for V-ellipticity are also simple: for � =(', ) ∈ V we find

Re �(�, �) = Re {−⟨ ,'⟩V + �1(', ) + �2( , )}

= Re {−⟨', ⟩V + ⟨', ⟩V + �2( , )}

= Re �2( , )

≥ �2∣ ∣2V

= �2(∣'∣2V + ∣ ∣2V )− �2∣'∣2V

≥ �2(∣'∣2V + ∣ ∣2V )− �2(∣'∣2V + ∣ ∣2H)

= �2∣�∣2V − �2∣�∣2ℋ.

We thus find that �(�, �) = ⟨A�, �⟩V,V∗ gives rise to the infinitesimalgenerator A of a C0 (indeed, analytic) semigroup on XE = ℋ. It is readilyargued that �(�, �) = ⟨−A�, �⟩ℋ for � ∈ D(A) = {� = (', ) ∈ ℋ∣ ∈ V =

24

H2L(0, l), A1'+A2 ∈ H, (EI'′′ + cDI

′′)(l) = 0, (EI'′′ + cDI ′′)′(l) = 0}

where

A =

(0 I−A1 −A2

)with A1' = ∂2(EI� ∂

2') and A2' = ∂2( cDI� ∂2').

Homework Exercises

∙ Ex. 10 : Some books define D(A) by

D(A) = (H4(0, l) ∩H2L(0, l))× (H4(0, l) ∩H2

L(0, l))

plus boundary conditions. We know A∣D(A) is an infinitesimal gen-erator of a C0 semigroup which, in turn, implies A∣D(A) is a closedoperator. You can show A∣D(A) is not closed. Therefore, we claim

that D(A) ∕= D(A) . Is this true? Look at both the damped andundamped cases.

25

13 Summary of Results on Analytic SemigroupGeneration by Sesquilinear Forms

We summarize results available for the special cases of analytic semigroupogeneration. For further details the reader can consult [BI, T].

Let V and H be complex Hilbert spaces with the Gelfand triple V ↪→H ↪→ V ∗. Let ⟨⋅, ⋅⟩V ∗,V be the duality product, and � : V × V → C be asesquilinear form such that � is

1. V continuous, i.e., ∣�(', )∣ ≤ ∣'∣V ∣ ∣V .

2. V -elliptic, i.e., Re �(',') ≥ �∣'∣2V . (We can, if necessary, replace thisby a shift: Re �(',') + �0∣'∣2H ≥ �∣'∣2V .)

As before, let A ∈ ℒ(V, V ∗) (note that this is −A in our old notation)and A : DA ⊂ H → H be defined such that

�(', ) = ⟨−A', ⟩V ∗,V for all ', ∈ V= ⟨−A', ⟩H ' ∈ DA, ∈ V.

Then we have ℛ(A) = V ∗, ℛ(A) = H, and 0 ∈ �(A). We can also note

Re �(',') = Re ⟨−A', '⟩V ∗,V ≥ �∣'∣2V .

for all ' ∈ V . In other words, Re⟨A', '⟩ ≤ −�∣'∣2V ≤ 0. Similarly, for' ∈ DA, Re ⟨A','⟩H ≤ 0 which implies A is dissipative. By Lumer Phillips,because A is dissipative and ℛ(A) = H, A is the infinitesimal generator ofa C0 semigroup of contractions S(t) on H.

We recall the definition of dissipativeness in a Banach space X. Anoperator B ∈ D ⊂ X → X is dissipative if for each x ∈ D(B) there existsx∗ ∈ F (x) ⊂ X∗ such that Re⟨x∗, Bx⟩X∗,X ≤ 0 where F (x) is the dualityset. Let’s apply this definition to X = V ∗, which is a reflexive Banachspace in its own right, with the operator B = A, A : V ⊂ V ∗ → V ∗. Wehave A being dissipative in the Banach space V ∗ means for x ∈ V thereexists x∗ ∈ F (X) ⊂ X∗ = V ∗∗ = V such that Re⟨x∗, Ax⟩V,V ∗ ≤ 0 or

Re⟨Ax, x∗⟩V ∗,V ≤ 0. However, we have this holding for every x∗ ∈ V ⊂ V ∗.(In particular, we can find such a x∗ in the duality set.) Therefore, A : V =D(A) ⊂ V ∗ → V ∗ is dissipative. Using Lumer Phillips again we have A is aninfinitesimal generator of a C0 semigroup of contractions S(t) on V ∗ whereS(t)∣H = S(t).

26

Recall DA = {x ∈ V ∣Ax ∈ H}. We defineˆDA = {x ∈ V ∣Ax ∈ V }

and define the operatorˆA = A∣ ˆ

DAin V . We have ℛ(

ˆA) = V ; therefore

the range statement needed for Lumer Phillips holds forˆA. However, for

ˆA to be dissipative in V , we must have for each x ∈ ˆ

DA ⊂ V there exists

x∗ ∈ F (x) ⊂ V ∗ such that Re⟨x∗, ˆAx⟩V ∗,V ≤ 0. We do not directly have

thatˆA is dissipative in V . To pursue this, we need to consider the Tanabe

estimates.

13.1 Tanabe Estimates(on “Regular Dissipative Operators”)

Suppose a sesquilinear form �(witℎassociatedoperatorA) is V continuousand V -elliptic. Then for Re� ≥ 0 and � ∕= 0, R�(A) = (�I − A)−1 ∈ℒ(V ∗, V ), and

1. ∣R�(A)'∣V ≤ 1� ∣'∣V ∗ for ' ∈ V ∗. (In other words, ∣R�(A)∣ℒ(V ∗,V ) ≤

1� .)

2. ∣R�(A)'∣H ≤ M0∣�∣ ∣'∣H for ' ∈ H where M0 = 1 +

� . (In other words,

∣R�(A)∣ℒ(H) ≤ M0∣�∣ .)

3. ∣R�(A)'∣V ∗ ≤ M0∣�∣ ∣'∣V ∗ for ' ∈ V ∗. (In other words, ∣R�(A)∣ℒ(V ∗) ≤

M0∣�∣ .)

4. ∣R�(A)'∣V ≤ M0∣�∣ ∣'∣V for ' ∈ V . (In other words, ∣R�(A)∣ℒ(V ) ≤ M0

∣�∣ .)

We give the arguments for 4. since it is a very strong and useful esti-mate. Define the dual or adjoint operator in the usual manner: defineA∗ ∈ ℒ(V, V ∗) by �(', ) = ⟨',−A∗ ⟩V,V ∗ for ', ∈ V . Then

�∗(', ) = �(', )

and�∗(', ) = ⟨',−A∗ ⟩V,V ∗

implies

⟨',−A∗ ⟩V,V ∗ = ⟨',−A∗ ⟩V,V ∗ .

27

Therefore, A∗ also satisfies the estimates 1.-3. above because �∗ is V contin-uous and V -elliptic. Applying 3 to A∗ gives us for Re� ≥ 0, � ∕= 0, ', ∈ V

∣⟨R�(A)', ⟩V,V ∗ ∣ = ∣⟨',R�(A∗) ⟩V,V ∗ ∣

≤ ∣'∣V ∣R�(A∗) ∣V ∗

≤ ∣'∣V M0∣�∣ ∣ ∣V ∗ .

Therefore, ∣R�(A)'∣V ≤ M0∣�∣ ∣'∣V .

We can show that the C0 semigroups from above are actually analytic.The theorem below gives a useful condition for analyticity.

Theorem 6 Let T (t) be a C0 semigroup on a Hilbert space X with infinites-imal generator A, with 0 ∈ �(A). Then a semigroup is analytic on X if thereexists a constant c such that

∣R�+i� (A)∣ℒ(X) ≤c

∣� ∣

for � > 0, � ∕= 0 where � = �+ i� .

See [P, Theorem II.5.2(b)].

From the Tanabe estimates, we have ∣R�(A)∣ℒ(X) ≤ c∣�∣ = c√

�2+�2≤ c∣� ∣

for X chosen as V,H, or V ∗. Therefore, our estimates suffice to provide an-alyticity of the associated semigroups. Thus we have the following theorem.

Theorem 7 Let V ↪→ H ↪→ V ∗ be a Gelfand triple. Assume the sesquilin-

ear form � is V continuous and V -elliptic. Let A, A, andˆA be defined as

above. Then

∙ A is an infinitesimal generator of an analytic semigroup S(t) of con-tractions on V ∗.

∙ A is an infinitesimal generator of an analytic semigroup S(t) of con-tractions on H.

∙ ˆA is an infinitesimal generator of an analytic semigroup

ˆS(t) of con-

tractions on V .

28

We also have

∙ domV ∗(A) = V .

∙ domH(A) = DA = {x ∈ V ∣Ax ∈ H}.

∙ domV (A) =ˆDA = {x ∈ V ∣Ax ∈ V }.

This is usually stated as A or A generate an analytic semigroup of contrac-tions on V ,H, V ∗.

Remark: The results of this section will be of fundamental importancein subsequent discussion on feedback control problems for infinite dimen-sional systems such as parabolic and strongly damped hyperbolic partialdifferential equations as well as functional differential equations. These prob-lems can be conveniently and profitably formulated in an abstract settingwith the systems defined in terms of Gelfand triples and where the asso-ciated algebraic Riccati equations for the feedback gains are formulated inan appropriately defined space V . In later sections we will discuss some ofthe results as developed in [BKcontrol, BI, 32] and summarized partially in[BSW].

13.2 Infinitesimal Generators in a General Banach Space

Recall that if A is an infinitesmal generator of a C0 semigroup T (t) in aHilbert space X, then S(t) = T ∗(t) is a C0 semigroup in X with infinitesimalgenerator A∗. Thus, if A is an infinitesimal generator, D(A) is dense in X.Similarly, if A∗ is an infinitesimal generator, D(A∗) is also dense in X. Wecan generalize this result in a general Banach space.

Theorem 8 If X is a reflexive Banach space and A is an infinitesimalgenerator of a C0 semigroup T (t) in X, then A∗ is an infinitesimal generatorof a C0 semigroup S(t) in X∗ and S(t) = T ∗(t) = (T (t))∗. In other words,(eA

∗t on X∗)∗ = eAt on X.

Corollary 1 If A is an infinitesimal generator of a C0 semigroup on V ∗,then A∗ is an infinitesimal generator on V ∗∗ = V for any reflexive Banachspace V .

29

In the formulations of the previous sections, we know A ∈ ℒ(V, V ∗) andA∗ ∈ ℒ(V ∗∗, V ∗) = ℒ(V, V ∗) are infinitesimal generators of C0 semigroups

of contractions on V ∗. In other words, S∗(t) = eA∗t is a C0 semigroup

of contractions on V ∗. Applying the previous corollary, we have (S∗(t)on V ∗)∗ = S(t) on V . However, S∗(t) ∈ ℒ(V ∗, V ∗) implies (S∗(t))∗ ∈ℒ(V ∗∗, V ∗∗) = ℒ(V, V ). Since V is a reflexive Hilbert space,

ˆS(t) defined

previously is exactly S(t)∣V (hereˆA = A∣ ˆ

DAis the infinitesimal generator of

ˆS(t) in V .

30

14 General Second Order Systems

14.1 Introduction to Second Order Systems

The ideas in Example 6 can be used to treat general second order systems.Consider the general abstract second order system

y(t) +A2y(t) +A1y(t) = f(t)

or, in variational form

⟨y(t), '⟩V ∗,V + �1(y(t), ') + �2(y(t), ') = ⟨f(t), '⟩V ∗,V (18)

where H is a complex Hilbert space. As usual, we assume that �1 and �2 aresesquilinear forms on V where V ↪→ H ↪→ V ∗ is a Gelfand triple. We also as-sume that �1 is continuous, V -elliptic and symmetric (�1(', ) = �1( ,')).We assume that �2 is continuous and satisfies a weakened ellipticity condi-tion which we formally call H-semiellipticity.

Definition 7 A sesquilinear form � on V is H-semielliptic if there is aconstant b ≥ 0 such that

Re �(',') ≥ b∣'∣2H for all v ∈ V.

Note that b = 0 is allowed in this definition.

Since �1 and �2 are continuous, we have that there exists Ai ∈ ℒ(V, V ∗),i = 1, 2, such that

�i(', ) = ⟨Ai', ⟩V ∗,V for all ', ∈ V, i = 1, 2.

Following the ideas of Example 6, we define spaces V = V × V andℋ = V × H and rewrite our second order system as a first order vectorsystem. Defining, for � = (', ), � = (g, ℎ) ∈ V, the sesquilinear form

�(�, �) = �((', ), (g, ℎ)) = −⟨ , g⟩V + �1(', ℎ) + �2( , ℎ),

we can write our system for x(t) = (y(t), y(t)) as

⟨x(t), �⟩ℋ + �(x(t), �) = ⟨F (t), �⟩ℋ � ∈ V

where F (t) = (0, f(t)). This is formally equivalent to the system

x(t) = Ax(t) + F (t)

31

where A is given by

D(A) = {x = (', ) ∈ ℋ∣ ∈ V and A1'+A2 ∈ H} (19)

and

A =

(0 I−A1 A2

). (20)

We first note that � is V continuous. To see this, we observe that �1 and�2 being V continuous implies

�1(', ℎ) ≤ 1∣'∣V ∣ℎ∣V

and�2(', ℎ) ≤ 2∣ ∣V ∣ℎ∣V .

We also have ∣�∣2V = ∣'∣2V + ∣ ∣2V and ∣�∣2V = ∣g∣2V + ∣ℎ∣2V . Putting all of thistogether, we have

∣�((', ), (g, ℎ))∣ ≤ ∣ ∣V ∣g∣V + 1∣'∣V ∣ℎ∣V + 2∣ ∣V ∣ℎ∣V≤ ∣�∣V ∣�∣V + 1∣�∣V ∣�∣V + 2∣�∣V ∣�∣V= (1 + 1 + 2)∣�∣V ∣�∣V .

This indeed implies that � is V continuous.As � is V continuous, A is the negative of the restriction to D(A) of the

operator A ∈ ℒ(V,V∗) defined by �(�, �) = ⟨A�, �⟩V∗,V so that �(�, �) =⟨−A�, �⟩ℋ for � ∈ D(A), � ∈ V.

14.2 Results for �2 V -elliptic

If both �1 and �2 are V -elliptic and �1 is the same as the V inner product,then we have exactly the case of Kelvin-Voigt damping in Example 6. Weproved with these assumptions, � is V-elliptic. ( Actually, we proved �(⋅, ⋅)+�0⟨⋅, ⋅⟩ℋ is V-elliptic.) Therefore, we have A is the infinitesimal generatorof an analytic semigroup (not of contractions since �0 > 0) on ℋ.

Even if the V inner product and �1 are not the same, this result is true.Since �1 is continuous, we have ∣�1(',')∣ ≤ 1∣'∣2V while �1 is symmetric

(i.e., �1(', ) = �1( ,')) implies Re �1(',') = �1(','). Thus, �1 beingV -elliptic is equivalent to �1 is V -coercive: �1(',') ≥ �∣'∣2V . Hence, �1 andthe inner product are equivalent. We may thus define V1 as the space V with�1 as inner product, obtaining a space that is setwise equal and topologicallyequivalent to V . In the space ℋ1 = V1×H the operator A is now associatedwith the V1 = V1 × V1-elliptic form �(1)(�, �) = ⟨−A�, �⟩ℋ1 that (as we

32

argued in Example 6) satisfies the conditions of our theorem. Hence, Agenerates an analytic semigroup in ℋ1 and hence an analytic semigroup inthe equivalent space ℋ. Thus we have

Theorem 9 Let V ↪→ H ↪→ V ∗ and suppose that �1 and �2 of (18) are Vcontinuous and V -elliptic sesquilinear forms on V and that �1 is symmetric.Then the operator A defined in (19) and (20) is the infinitesimal generatorof an analytic semigroup in ℋ = V ×H.

14.3 Results for �2 H-semielliptic

If �2 is not V -elliptic, then we will not, in general, obtain an analytic solutionsemigroup for our system. We will obtain a C0 semigroup, but must work alittle more to obtain such. So assume that �2 is only H-semielliptic. Thenwe have A defined in (19) and (20) is dissipative in ℋ1 since

Re ⟨Ax, x⟩ℋ1 = Re {�1( ,')− �1(', )− �2( , )}

= Re {�1(', )− �1(', )− �2( , )}

= −Re �2( , )

≤ −b∣ ∣2H ≤ 0.

To argue that A is a generator, we use the Lumer Phillips theorem; thus weneed to argue that for some � > 0, the range of �I −A is ℋ1. Thus, given� = (g, ℎ) ∈ ℋ1, we wish to solve (�−A)� = � for � = (', ) ∈ D(A).

So we consider the equation

(�−A)(', ) = (g, ℎ) for (g, ℎ) ∈ V1 ×H.

This is equivalent to {�'− = g

� +A1'+A2 = ℎ.(21)

If we formally solve the first equation for = �' − g and substitute thisinto the second equation, we obtain

�2'− �g +A1'+A2(�'− g) = ℎ

or�2'+A1'+ �A2' = ℎ+ �g +A2g. (22)

33

This equation must be solved for ' ∈ V1 ( and then defined by = �'−gwill also be in V1).

These formal calculations suggest that we define for � > 0 the associatedsesquilinear form on V × V → C

��(', ) = �2⟨', ⟩H + �1(', ) + ��2(', ).

Since �1 is V -elliptic and �2 is H-semielliptic we have

Re ��(',') = �2∣'∣2H + Re �1(',') + � Re �2(',')

≥ �2∣'∣2H + c1∣'∣2V + �b∣'∣2H

= �(�+ b)∣'∣2H + c1∣'∣2V

> c1∣'∣2V

for � = �(� + b) > 0. Hence �� is V -elliptic and (22) is solvable for ' ∈ Vby Lax-Milgram. It follows that (21) is solvable for (', ) ∈ D(A), i.e.,ℛ(�−A) = ℋ1. Thus we have that A generates a contraction semigroup inℋ1 and a C0 semigroup in ℋ.

Theorem 10 Let V ↪→ H ↪→ V ∗ and suppose that �1 and �2 of (18) satisfy:�1 is V -elliptic, V continuous and symmetric, �2 is V continuous and H-semielliptic. Then A defined by (19) and (20) generates a C0 semigroup inℋ.

14.4 Stronger Assumptions for �2

If we strengthen the assumption on the damping form, we can obtain astronger result.

Theorem 11 Suppose �1 is V -elliptic, V continuous, and symmetric and�2 is H-elliptic, V continuous, and symmetric. Then A is the infinitesimalgenerator of a C0 semigroup T (t) in ℋ = V ×H that is exponentially stable,i.e., ∣T (t)�∣ℋ ≤Me−!t∣�∣ℋ for some ! > 0.

To motivate the arguments used to establish this result, we consider for! > 0 the change of dependent variable y(t) = e−!tr(t) in the equation

y +A2y(t) +A1y(t) = 0. (23)

34

Upon substitution, we obtain

r(t) + A2r(t) + A1r(t) = 0 (24)

whereA1 = A1 − !A2 + !2I

A2 = A2 − 2!I.

This suggests that we define the sesquilinear forms

�1(', ) = �1(', )− !�2(', ) + !2⟨', ⟩H

�2(', ) = �2(', )− 2!⟨', ⟩H

so that �i(', ) = ⟨Ai', ⟩V ∗,V , i = 1, 2, and the transformed variationalform of (23) is

⟨r(t), '⟩V ∗,V + �1(r(t), ') + �2(r(t), ') = 0

for ' ∈ V .We observe that �1, �2 are continuous and �1 is symmetric since both �1

and �2 are. Since �2 is symmetric (hence �2(',') is real) and continuouswith �2(',') ≤ k2∣'∣2V , we have for ' ∈ V

Re �1(',') = �1(',')= �1(',')− !�2(',') + !2∣'∣2H≥ c1∣'∣2V − ! 2∣'∣2V + !2∣'∣2H≥ (c1 − ! 2)∣'∣2V .

Hence �1 is V -elliptic if ! > 0 is chosen so that ! < c1 2

.Moreover, we find that �2 is H-semielliptic if ! is chosen properly since

Re �2(',') = Re �2(',')− 2!∣'∣2H ≥ (b− 2!)∣'∣2H .

Therefore, �2 is H-semielliptic if ! < b2 .

Thus, if we choose ! > 0 as ! = 12 min { b2 ,

c1 2}, we find that �1 and �2

satisfy the assumptions of Theorem 10. By the arguments preceding thattheorem, we see that

A =

(0 I

−A1 −A2

)(see (19) and (20)) generates a contraction semigroup T (t) on ℋ1 = V1×Hwhere V1 is V taken with �1 as inner product (V1 is equivalent to V ).

35

Now let T (t) be the C0-semigroup generated by A (see (19), (20) and

Theorem 10). If x(t) =

(y(t)y(t)

)and w(t) =

(r(t)r(t)

)are solutions of

(23) and (24) respectively, we have x(t) = T (t)x0 where x0 =

(y0

w0

)and

w(t) = T (t)w0. Since y(t) = e−wtr(t) and y(t) = −wewtr(t) + ewtr(t), wesee that x(t) = ewtΓw(t) where

Γ =

(1 0−w 1

)and w0 = Γ−1x0. It follows since ∣T (t)∣ℋ1

≤ 1 that

∣T (t)x0∣ℋ1≤ e−wt∣ΓT (t)Γ−1x0∣ℋ1

≤ Me−wt∣x0∣ℋ1.

Since ℋ1 and ℋ = V ×H are norm equivalent, we thus find that the semi-group T (t) is exponentially stable in ℋ.

36

15 Abstract Cauchy Problem

It is of great practical as well as theoretical interest to know when, and inwhat sense, solutions of the abstract equations

x(t) = Ax(t) + f(t)

x(0) = x0

(25)

exist. Moreover, representations of such solutions in terms of a variation ofparameters formula and the semigroup generated by A will play a fundamen-tal role in control and estimation formulations. We begin by summarizingresults available in the standard literature on linear semigroups and abstractCauchy problems.

Consider the abstract Cauchy problem (ACP) given by (25) where A isthe infinitesimal generator of a C0-semigroup T (t) in a Hilbert space H. Wedefine a mild solution xm of (25) as functions given by

xm(t) = T (t)x0 +

∫ t

0T (t− s)f(s)ds (26)

whenever this entity is well defined (i.e., f is sufficiently smooth).We say that x : [0, T ] → H is a strong solution of (ACP) if x ∈

C([0, T ], H)∩C1((0, T ], H), x(t) ∈ D(A) for t ∈ (0, T ], and x satisfies (25)

on [0, T ].We have the following series of results from the literature.

Theorem 12 If f ∈ L1((0, T ), H) and x0 ∈ H, there is at most one strongsolution of (25). If a strong solution exists, it is given by (26).

Theorem 13 If x0 ∈ D(A) and f ∈ C1([0, T ], H), then xm given by (26)provides the unique strong solution of (25).

Theorem 14 If x0 ∈ D(A), f ∈ C([0, T ], H), f(t) ∈ D(A) for each t ∈[0, T ] and Af ∈ C([0, T ], H), then (26) provides the unique strong solutionof (25).

Theorem 15 Suppose A is the infinitesimal generator of an analytic semi-group T (t) on H. Then if x0 ∈ H and f is Holder continuous (i.e. ∣f(t)−f(s)∣ ≤ k∣t−s∣ for some ≤ 1), then xm of (26) provides the unique strongsolution of (25).

37

Unfortunately, all of these powerful results are too restrictive for use inmany applications, including control theory, where typically f(t) = Bu(t) isnot continuous, let alone Holder continuous or C1. For this reason, a weakerformulation is more appropriate. For this, we follow the presentations ofLions, Wolka, and Tanabe which are developed in the context of sesquilinearforms and Gelfand triples, V ↪→ H ↪→ V ∗, where V,H, V ∗ are Hilbert spaces.

We define the solution space W(0, T ) by

W(0, T ) = {g ∈ L2((0, T ), V ) :dg

dt∈ L2((0, T ), V ∗)}

with scalar product

⟨g, ℎ⟩W =

∫ T

0⟨g(t), ℎ(t)⟩V dt+

∫ T

0⟨dgdt

(t),dℎ

dt(t)⟩V ∗dt.

Then it can be shown that W(0, T ) is a Hilbert space which embeds contin-uously into C([0, T ], H).

Assume � : V × V → C satisfies for ', ∈ V

Re �(',') ≥ c1∣'∣2V − �0∣'∣2H c1 ≥ 0, �0 real, for all ' ∈ V,

∣�(', )∣ ≤ ∣'∣V ∣ ∣V for all ', ∈ V.

Then, as already discussed, we have A ∈ ℒ(V, V ∗) such that �(', ) =⟨A', ⟩V ∗,V = ⟨−A', ⟩H where A is the densely defined restriction of −Ato the set DA = {' ∈ V ∣A' ∈ H}. We have moreover, that A is theinfinitesimal generator of an analytic semigroup T (t) on H. In fact, fromTheorem 8 we have that −A is the generator of an analytic semigroup T (t)in V,H and V ∗ and T (t) agrees with T (t) on V and H.

We may consider solutions of (25) in the sense of V ∗, i.e., in the sense

⟨x(t), ⟩V ∗,V + �(x(t), ) = ⟨f(t), ⟩V ∗,V for ∈ V,x(0) = x0.

(27)

By a strong solution of (25) in the V ∗ sense (also quite frequently calleda weak or variational or distributional solution), we shall mean a functionx ∈ L2((0, T ), V ) such that x ∈ L2((0, T ), V ∗) and (27) (or equivalentlyx(t) + Ax(t) = f(t)) holds almost everywhere on (0, T ). Similarily, mildsolutions xm ∈ V ∗ are given by the analogue of (26)

xm(t) = T (t)x0 +

∫ t

0T (t− s)f(s)ds. (28)

We then have the fundamental existence and uniqueness theorem.

38

Theorem 16 Suppose x0 ∈ H and f ∈ L2((0, T ), V ∗). Then (27) has aunique strong solution in the V ∗ or variational sense and this is given bythe mild solution (28).

Proof

Let {�i}∞1 ⊂ V be a linearly independent total subset (i.e., a basis) of V .

We define the “Galerkin” approximations by xk(t) =k∑i=1

wi(t)'i where the

coefficients {wi} are chosen so that

⟨xk(t), 'j⟩H + �(xk(t), 'j) = ⟨f(t), 'j⟩V ∗,V (29)

for j = 1, . . . , k, satisfying the intial condition

xk(0) = xk0

where

xk0 =

k∑i=1

wi0'i → x0

in H as k →∞. Equivalently, (29) can be written as

k∑i=1

wi(t)⟨'i, 'j⟩+k∑i=1

wi(t)�('i, 'j) = Fj(t)

where Fj(t) = ⟨f(t), 'j⟩V ∗,V for j = 1, . . . , k. Therefore, w1, . . . , wk areunique solutions to a vector ordinary differential equation system.

Multiplying (29) by wj and summing over j = 1, . . . , k, we obtain

⟨xk(t), xk(t)⟩H + �(xk(t), xk(t)) = ⟨f(t), xk(t)⟩V ∗,V

with xk(0) = xk0 → x0 in H. Therefore

1

2

d

dt∣xk(t)∣2H + �(xk(t), xk(t)) = ⟨f(t), xk(t)⟩V ∗,V . (30)

Integrating (30), we obtain

1

2∣xk(t)∣2H −

1

2∣xk(0)∣2H +

∫ t

0�(xk(s), xk(s))ds =

∫ t

0⟨f(s), xk(s)⟩V ∗,V ds.

Using the fact that � is V -elliptic, we have

39

1

2∣xk(t)∣2H + c1

∫ t

0∣xk(s)∣2V ds ≤

1

2∣xk(0)∣2H +

∫ t

0∣⟨f(s), xk(s)⟩V ∗,V ∣ds

≤ 1

2∣xk(0)∣2H +

∫ t

0(

1

4�∣f(s)∣2V ∗ + �∣xk(s)∣2V )ds.

Therefore,

1

2∣xk(t)∣2H + (c1 − �)

∫ t

0∣xk(s)∣2V ds ≤

1

2∣xk(0)∣2H +

∫ t

0

1

4�∣f(s)∣2V ∗ds (31)

or

1

2∣xk(t)∣2H + (c1 − �)

∫ t

0∣xk(s)∣2V ds ≤

1

2∣xk(0)∣2H +

1

4�∣f ∣2L2((0,t),V ∗).

This implies we have {xk} bounded in C((0, T ), H) and in L2((0, T ), V ).Since L2((0, T ), V ) is a Hilbert space, we can choose {xkn ∣xkn ⇀ x ∈L2((0, T ), V )} to be a convergent subsequence of xk. Without loss of gener-ality, we reindex and denote xkn by xk. Then the limit x is our candidatefor a solution where xk ⇀ x in L2((0, T ), V ).

Let �(t) ∈ C1(0, T ) with �(T ) = 0 and �(0) = 0 and define Ψj(t, ⋅) byΨj(t, ⋅) = �(t)'j . Multiplying (29) by �(t) and integrating, we have∫ T

0(⟨xk(t), 'j⟩H�(t) + �(xk(t), 'j)�(t)− ⟨f(t), 'j⟩V ∗,V �(t)) dt = 0. (32)

Integrating by parts, we find that (32) becomes

−∫ T

0⟨xk(t), 'j⟩�(t)dt+

∫ T

0�(xk(t), 'j)�(t)−

∫ T

0⟨f(t), 'j⟩V ∗,V �(t)dt = 0.

We can now let k → ∞ and pass the limit through term by term toobtain∫ T

0−⟨x(t), 'j⟩�(t)dt+

∫ T

0�(x(t), 'j)�(t)dt−

∫ T

0⟨f(t), 'j⟩V ∗,V �(t)dt = 0,

(33)holding for all 'j ∈ V . Recall that {'j} is a total subset of V and observethat the �s such as above are dense in L2(0, T ). Thus we have (33) holdingfor all Ψ = '� in L2((0, T ), V ). We can rewrite �(x(t), ')�(t) as Ax(t)Ψ

and∫ T

0 ⟨f(t), '⟩V ∗,V �(t)dt as f(Ψ).

40

Therefore, (33) becomes

⟨ ddtx,Ψ⟩V ∗,V + (Ax− f)Ψ = 0

where Ψ ∈ L2((0, T ), V ), or x satisfies dxdt +Ax− f = 0 in the L2((0, T ), V )∗

sense. However, we have the following needed theorem (details can be foundin [E]).

Theorem 17 Let X be a reflexive Banach space. Then

Lp((0, T ), X)∗ ∼= Lq((0, T ), X∗)

where 1p + 1

q = 1, 1 < p <∞.

Returning to our arguments we thus have that the solution to the equa-tion exists in the L2((0, T ), V )∗ = L2((0, T ), V ∗) sense and is given byx. To obtain x(0) = x0, we may use the same arguments with arbitrary� ∈ C1(0, T ), �(T ) = 0, but �(0) ∕= 0.

To prove uniqueness of the solution, it suffices to argue that the solutioncorresponding to x0 = 0, f = 0 is identically zero. With these specific valuesfor f and x0, (27) can be written as

⟨x(t), '⟩V ∗,V + �(x(t), ') = 0, (34)

x(0) = 0.

Let ' = x(t). Then (34) becomes

1

2

d

dt∣x(t)∣2H + �(x(t), x(t)) = 0.

Integrating by parts and using the V -ellipticity of �, we obtain

1

2∣x(t)∣2H +

∫ t

0c1∣x(s)∣2V ds ≤ 0.

Therefore, x(t) = 0 and thus the solution is unique.To estsblish continuous dependence of the solution, define

x(⋅;x0, f) : (x0, f) ∈ H×L2((0, T ), V ∗)→ x ∈ L2((0, T ), V )∩C((0, T ), H).

Therefore, x ∈ L2((0, T ), V ∗). Taking the limits in (31) and using the prop-erty that the norms are weakly lower semi-continuous, we obtain the follow-ing relation:

1

2∣x(t)∣2H + (c1 − �)

∫ t

0∣x(s)∣2V ds ≤

1

2∣x0∣2 +

1

4�

∫ t

0∣f(s)∣2V ∗ds.

41

and hence

supt∈[0,T ]

1

2∣x(t)∣2H + (c1 − �)

∫ T

0∣x(s)∣2V ds ≤

1

2∣x0∣2H +

1

4�

∫ T

0∣f(s)∣2V ∗ds.

Since the map (x0, f) → x is a linear map on H × L2((0, T ), V ∗), we haveimmediately that x is continuous on H × L2((0, T ), V ∗) into C((0, T ), H),L2((0, T ), V ∗) and L2((0, T ), V ).

Finally, we need to prove the equivalence between this solution and themild solution given by (28). From (28) we have that the map (x0, f) →x(⋅, x0, f) is continuous from H × L2((0, T ), V ∗) to L2((0, T ), V ∗). We havejust argued that the variational solution xvar(⋅, x0, f) is continuous in thesame sense. Thus xvar and xm are both continuous in the above sense.Recall that if two functions agree on a dense subset of some set, then thesolutions must agree on the entire set. Therefore, if there is a dense subsetof H × L2((0, T ), V ∗) on which xvar and xm agree, then they will agree onthe entire set.

Choose x0 ∈ DA and f ∈ C1((0, T ), H). Then Theorem 13 guaran-tees that xm is the unique solution in the H sense. However, if xm is astrong solution in the H sense, then it must also be a variational solution(i.e., a strong solution in the V ∗ sense). However, the mild solution beingunique means xm(⋅, x0, f) = xvar(⋅, x0, f) for (x0, f) ∈ DA × C1((0, T ), H).But DA × C1((0, T ), H) is dense in H × L2((0, T ), V ∗). Hence, we havethe equivalence between the solutions corresponding to data (x0, f) in H ×L2((0, T ), V ∗).

42

16 Weak Formulations for Second Order Systems

16.1 Model Formulation

We return to the general second order systems (18) of Section 14 to illustratewell-posedness ideas in the context of the abstract hyperbolic model withtime dependent stiffness and damping given by

⟨y(t), ⟩V ∗,V + d(t; y(t), ) + a(t; y(t), ) = ⟨f(t), ⟩V ∗,V

where V ⊂ VD ⊂ H ⊂ V ∗D ⊂ V ∗ are Hilbert spaces with continuous anddense injections, where H is identified with its dual and ⟨⋅, ⋅⟩ denotes theassociated duality product. We first show under reasonable assumptions onthe time-dependent sesquilinear forms a(t; ⋅, ⋅) : V × V → C and d(t; ⋅, ⋅) :VD × VD → C that this model possesses a unique solution and that thesolution depends continuously on the data of the problem. We also considerwell-posedness as well as finite element type approximations in associatedinverse problems. The model is a weak formulation that includes modelsin abstract differential operator form that include plate, beam and shellequations with several important kinds of damping.

Applications for such systems are abundant and range from systemswith periodic or structured pattern time dependence that occurs for exam-ple in thermally dependent systems orbiting in space (periodic exposure tosunlight) to earth bound structures (bridges, buildings) and aircraft/spacestructure components with extreme temperature exposures (winter vs. sum-mer). On a longer time scale, such systems are important in time dependenthealth of elastic structures where long term (slowly varying) time depen-dence of parameters may be used in detecting aging/fatiguing as repre-sented by changes in stiffness and/or damping. Moreover, applications maybe found in modern “smart material structures” [BSW, RCS] where one“controls” damping and/or elasticity via piezoceramic patches, electricallyactive polymers, etc.

As before let V and H be complex Hilbert spaces forming a “Gelfandtriple” V ⊂ H = H∗ ⊂ V ∗ with duality product ⟨⋅, ⋅⟩V ∗,V . The injectionsare assumed to be dense and continuous and the spaces are assumed to beseparable. Moreover, we assume that there exists a Hilbert space VD (thedamping space), such that V ⊂ VD ⊂ H = H∗ ⊂ V ∗D ⊂ V ∗, allowing fora wide class of damping models. Thus the duality products ⟨⋅, ⋅⟩V ∗,V and⟨⋅, ⋅⟩V ∗D,VD are the natural extensions by continuity of the inner product ⟨⋅, ⋅⟩in H to V ∗ × V and V ∗D × VD, respectively. The H-norm is denoted by ∣ ⋅ ∣or ∣ ⋅ ∣H for clarity if needed and the V and VD-norms are denoted ∣ ⋅ ∣V and

43

∣ ⋅ ∣VD , respectively. We denote ∂ty by y, where y = y(t) is considered as afunction of time taking values in one of the occurring Hilbert spaces, thatis, y(t, �) = y(t)(�) where y(t) ∈ H, for example.

Our goal is to investigate the system

y(t) +D(t)y(t) +A(t)y(t) = f(t), in V ∗, t ∈ (0, T ); (35)

with y(0) = y0 ∈ V and y(0) = y1 ∈ H which is equivalent to

⟨y(t), ⟩V ∗,V + ⟨D(t)y(t), ⟩V ∗D,VD + ⟨A(t)y(t), ⟩V ∗,V = ⟨f(t), ⟩V ∗,V , (36)

for all ∈ V and t ∈ (0, T ), with y(0) = y0 ∈ V and y(0) = y1 ∈ H.Following standard terminology we will call a function y ∈ L2(0, T ;V ),

with y ∈ L2(0, T ;H) and y ∈ L2(0, T ;V ∗) a weak solution of the initial-value problem (35) if it solves the equation (36) in the strong V ∗ sense ofthe previous section, or, equivalently, solves the equation (45) below, withy(0) = y0 ∈ V and y(0) = y1 ∈ H given.

We will assume that the operators A and D arise (as defined precisely in(42)-(43) below) from time-dependent sesquilinear forms a and d satisfyingthe following natural ellipticity, coercivity and differentiability conditions.

First, we assume hermitian symmetry, that is

(H1) a(t, �, ) = a(t, , �) for all �, ∈ V .

(H2) ∣a(t;�, )∣ ≤ c1∣�∣V ∣ ∣V , �, ∈ V where c1 is independent of t.

We assume, further, that

(H3) a(t;�, ) for �, ∈ V fixed is continuously differentiable with respectto t for t ∈ [0, T ] (T finite) and

∣a(t;�, )∣ ≤ c2∣�∣V ∣ ∣V , for all t ∈ [0, T ], (37)

c2 once again independent of t.

We also assume that the sesquilinear form a(t;�, ) is V -elliptic, so that

(H4) There exist a constant � > 0 such that

∣a(t;�, �)∣ ≥ �∣�∣2V for all t ∈ [0, T ], and for all u ∈ V. (38)

For the sesquilinear form d we assume similarly

44

(H5) ∣d(t;�, )∣ ≤ c3∣�∣VD ∣ ∣VD , �, ∈ VD where c3 is independent of t.

We assume, further, that

(H6) d(t;�, ) for �, ∈ VD fixed is continuously differentiable with respectto t for t ∈ [0, T ] (T finite) and

∣d(t;�, )∣ ≤ c4∣�∣VD ∣ ∣VD , for all t ∈ [0, T ], (39)

c4 once again independent of t.

Then t → d(t;�, ) and t → a(t;�, ) are C1[0, T ] for all �, ∈ VD, and�, ∈ V , respectively, which implies that d(t;�, ) and a(t;�, ) are suf-ficiently well-behaved in order to have existence for (35) or (36). We alsoassume that the sesquilinear form d(t;�, ) is VD-coercive. That is,

(H7) There exist constants �d and �d > 0, such that

Re d(t;�, �) + �d∣�∣2 ≥ �d∣�∣2VD for all t ∈ [0, T ], and for all � ∈ VD.(40)

We know then from our earlier considerations that there exist represen-tation operators A(t) and D(t)

A(t) : V → V ∗, D(t) : VD → V ∗D, (41)

which for each fixed t are continuous and linear, with

a(t;�, ) = ⟨A(t)�, ⟩V ∗,V , for all �, ∈ V, (42)

andd(t;�, ) = ⟨D(t)�, ⟩V ∗D,VD , for all �, ∈ VD. (43)

We will now consider the following problem: Given finite T and f ∈L2(0, T ;V ∗D) along with initial conditions

y0 ∈ V , y1 ∈ H,

we wish to find a function y ∈ L2(0, T ;V ), y ∈ L2(0, T ;VD) such that in V ∗

we have {y(t) +D(t)y(t) +A(t)y(t) = f(t), t ∈ (0, T ),

y(0) = y0, y(0) = y1.(44)

45

That is, for f ∈ L2(0, T ;V ∗D)

⟨y(t), ⟩V ∗,V + d(t; y(t), ) + a(t; y(t), ) = ⟨f(t), ⟩V ∗,V for all ∈ V.(45)

We remark that (45) makes sense since f(t) ∈ V ∗D ⊂ V ∗.This formulation covers linear beam (i.e., Example 6 discussed earlier),

plate and shell models with numerous damping models (Kelvin-Voigt, vis-cous, square-root, structural and spatial hysteresis) frequently studied inthe literature. The formulation above is non-standard in the sense that thedamping sesquilinear form is incorporated in the variational model and istime-dependent. The problem without damping (d = 0) and f ∈ L2(0, T ;H)was treated by Lions in [Li], and subsequently in [W]. The less general casewithout damping and V = H1

0 (Ω) is treated for example in [Ev]. The modelabove with d : VD → C independent of time is treated in [BSW]. Thefollowing theorem is a time-dependent extension of the previous results.

Theorem 18 Assume that (f, y0, y1) ∈ L2(0, T ;V ∗D) × V × H. Then thereexists a unique solution y to (45) with (y, y) ∈ L2(0, T ;V ) × L2(0, T ;VD),and the mapping

(f, y0, y1)→ (y, y), (46)

is continuous and linear on

L2(0, T ;V ∗D)× V ×H → L2(0, T ;V )× L2(0, T ;VD). (47)

As we will see, Theorem 18 can then be extended to stronger smoothnessresults on solutions.

Theorem 19 Assume that (f, y0, y1) ∈ L2(0, T ;V ∗D) × V × H. Then thereexists (perhaps after modifications on a set of measure zero) a unique solu-tion y to (45) with (y, y) ∈ C(0, T ;V )× (C(0, T ;H)∩L2(0, T ;VD)), and themapping

(f, y0, y1)→ (y, y), (48)

is continuous and linear on

L2(0, T ;V ∗D)× V ×H → C(0, T ;V )× (C(0, T ;H) ∩ L2(0, T ;VD)). (49)

Remark: If we only have that the inequality (40) for the dampingsesquilinear form d is satisfied with �d = 0, the results are still true withmodifications. Then it will be necessary that f ∈ L2(0, T ;H) and oneobtains only that y ∈ L2(0, T ;H); that is, we have the same results as if

46

there was no damping. (See, e.g., [Li].) As an added comment, we note thatJ. L. Lions was one of the early and most prolific contributors to functionalanalytic formulations of partial differential equations and control theory. Hismany contributions include the seminal texts [Li, LiMag].

16.2 Discussion of the Model

It is well known from the literature that the strong form of the operatorformulation (35) of the problem in general causes computational problemsdue to irregularities stemming from non-smooth terms - typically in theforce/moment terms in, for example, elasticity problems. The weak for-mulation has proven advantageous both for theoretical and practical pur-poses, specifically in the effort to estimate parameters or for control purposes[BSW]. To give a particular example illustrating and motivating our dis-cussions here, we consider a version of the beam of Example 6 but withboth ends fixed. For an Euler-Bernoulli beam of length l, width b, thicknessℎ and linear density �, where the parameters b, ℎ, � may be functions thatdepend on time and/or spatial position � along the beam, the equation fortransverse displacements y = y(t, �) (in strong form [BSW]) is given by

�∂2y

∂t2+

∂2

∂�2

{CDI

∂3y

∂�2∂t+ EI

∂2y

∂�2

}= f 0 < � < l, (50)

with fixed end boundary conditions

y(t, 0) =∂y

∂�(t, 0) = y(t, l) =

∂y

∂�(t, l). (51)

Here we assume Kevin-Voigt structural damping with damping coefficientCDI = CDI(t, �) and the possibly time and spatially dependent stiffness

coefficient given by EI = EI(t, �). For simplicity we assume � is constant

and scale the system by taking � = 1. One can readily compute that EI =Eℎ3b/12, CDI = CDℎ

3b/12, where the Young’s modulus E, the dampingcoefficient CD and the geometric parameters ℎ, b may in general all be timeand/or spatially dependent. As we have seen, in weak form this can bewritten

⟨y(t), ⟩V ∗,V + ⟨CDI∂y2(t)

∂�2+ EI

∂2y(t)

∂�2,∂2

∂�2⟩H = ⟨f(t), ⟩V ∗,V , (52)

for all ∈ V , where H = L2(0, l) and V = VD = H20 (0, l) with

H20 (0, l) ≡ { ∈ H2(0, l)∣ (0) = ′(0) = (l) = ′(l) = 0}. (53)

47

Here we have adopted the usual notation ′ = ∂ ∂� .

This has the form (36) or (45) with

a(t;�, ) = ⟨A(t)�, ⟩V ∗,V =

∫ l

0EI(t, �)�′′(�) ′′(�)d� (54)

d(t;�, ) = ⟨D(t)�, ⟩V ∗,V =

∫ l

0CDI(t, �)�′′(�) ′′(�)d�. (55)

Models such as this can be generalized to higher dimensions (in Rn forn = 2, 3) to treat more general beams, plates, shells, and solid bodies [BSW,RCS]. Of great interest are a number of useful damping models that can bereadily used with these equations and treated using the abstract formulationdeveloped here. We discuss briefly some of these damping models withoutgoing into much detail, as the purpose of this section is to establish well-posedness and approximation properties of the abstract model. We considerbriefly several damping models of interest in practice.

Time-dependent Kelvin-Voigt damping: Let ! ⊂ [0, l], with 1! denotingthe characteristic function of !. Let , � > 0 denote material parameters andlet t→ k(t) denote a sufficiently smooth function. Then a time−dependentdamping sesquilinear form is given by

d(t;�, ) =

∫ l

0( + �k(t)1!(�)�′′(�) ′′(�)d�, (56)

for �, ∈ VD = V = H20 (0, l). This gives a model for a mechanical structure

damped by a time-varying actuator, localized somewhere inside the struc-ture. This could be piezoceramic actuators or other “smart” devices, withthe possibility of them varying in time.

Time-dependent viscous damping: This is a velocity-proportional damp-ing, given (with the notation from above) by the sesquilinear form

d(t;�, ) =

∫ l

0k(t, �)�(�) (�)d�, (57)

with k ∈ C1(0, T ;L∞(0, l)) denoting the damping coefficient. One can takeVD = L2(0, l) here.

Time-dependent spatial hysteresis damping: This model, without time-dependence, is discussed in [DLR], and, as noted in [BSW], it has been

48

shown to be appropriate for composite material models where graphite fibersare embedded in an epoxy matrix. The time-dependent sesquilinear formthat we consider here can now be constructed with the following compactoperator K(t) on L2(0, l):

(K(t)�)(�) =

∫ l

0k(t, �, �)�(�)d�, (58)

where the nonnegative integral kernel k belongs to C1(0, T ;L∞(0, l)×L∞(0, l)).Letting

�(�) =

∫ l

0�(�, �)d� (59)

denote some material property, we can define

d(t;�, ) =

∫ l

0(�(�)−K(t))�′(�) ′(�)d�, (60)

with VD = H1(0, l).

16.3 Proof of Theorems 18 and 19

As with the first order systems investigated earlier, we will follow a standardGalerkin approximation method (see for example [BSW, LiMag, W]) withnecessary, non-trivial modifications as given in [BaPed] due to the presenceof the time-dependent forms. So, let {wj}∞1 denote a basis (i.e., a linearlyindependent total set) in V that is also a basis in V . This is possible sinceV is dense in H. For a fixed m we denote by Vm the finite dimensionalsubspace spanned by {wj}m1 , and we let u0

m and y1m be chosen in Vm such

thaty0m → y0 in V , y1

m → y1 in H, for m→∞. (61)

We now define the approximate solution ym(t) of order m of our problemin the following way:

ym(t) =m∑j=1

gjm(t)wj , (62)

where the gjm(t) are determined uniquely from the m-dimensional linearsystem:

(ym(t), wj)+d(t; ym(t), wj)+a(t; ym(t), wj) = ⟨f(t), wj⟩V ∗,V , j = 1, 2, ...,m;(63)

49

with ym(0) = y0m and ym(0) = y1

m. Multiplying (63) with gjm(t) and sum-ming over j yields

(ym(t), ym(t)) + d(t; ym(t), ym(t)) + a(t; ym(t), ym(t)) = ⟨f(t), ym(t)⟩V ∗,V .(64)

Now, since

d

dta(t; ym(t), ym(t)) = 2 Re a(t; ym(t), ym(t)) + a(t; ym(t), ym(t)), (65)

we see that

d

dt{∣ym(t)∣2 + a(t; ym(t), ym(t))}+ 2 Re d(t; ym(t), ym(t)) =

a(t; ym(t), ym(t)) + 2 Re⟨f(t), ym(t)⟩V ∗,V

and by integrating this equality we find

∣ym(t)∣2 + a(t; ym(t), ym(t)) +

∫ t

02 Re d(t; ym(s), ym(s))ds =

∣ym(0)∣2 + a(0; y0m, y

0m) +

∫ t

0a(s; ym(s), ym(s))ds+

∫ t

02 Re⟨f(s), ym(s)⟩V ∗,V ds.

Using the coercivity conditions for a and d, together with the inequality(recall that f(s) ∈ V ∗D)

∣⟨f(s), ym(s)⟩V ∗,V ∣ ≤1

4�∣f(s)∣2V ∗D + �∣ym(s)∣2VD (66)

we obtain, for all � > 0

∣ym(t)∣2 + �∣ym(t)∣2V +

∫ t

02(�d − �)∣ym(s)∣2VDds ≤ ∣y

1m∣2 + c1∣y0

m∣2V +

c2

∫ t

0∣ym(s)∣2V ds+ 2�d

∫ t

0∣ym(s)∣2ds+

∫ t

0

1

2�∣f(s)∣2V ∗Dds.

Since y0m → y0 in V , y1

m → y1 in H and f ∈ L2(0, T ;V ∗D), we have that, for� > 0 fixed and m large, there exist a constant C > 0, such that

∣y1m∣2 + c1∣y0

m∣2V +

∫ t

0

1

2�∣f(s)∣2V ∗Dds ≤ C, (67)

hence

∣ym(t)∣2 + �∣ym(t)∣2V +

∫ t

02(�d − �)∣ym(s)∣2VDds ≤

C + c2

∫ t

0∣ym(s)∣2V ds+ 2�d

∫ t

0∣ym(s)∣2ds.

50

Then, in particular

∣ym(t)∣2 + �∣ym(t)∣2V ≤

C + c2

∫ t

0∣ym(s)∣2V ds+ 2�d

∫ t

0∣ym(s)∣2ds. (68)

By Gronwall’s inequality we then see that the sequence {ym} is bounded inC(0, T ;H) and that the sequence {ym} is bounded in C(0, T ;V ). From thisfact together with the inequality (68) we conclude that {ym} is also boundedin L2(0, T ;VD). Then it is possible to extract a subsequence {ymk} ⊂ {ym}and functions y ∈ L2(0, T ;V ) and y ∈ L2(0, T ;VD), such that ymk ⇀ y,weakly in L2(0, T ;V ) and ymk ⇀ y, weakly in L2(0, T ;VD). But for 0 ≤ t <T we have in V , hence in VD and H, that

ymk(t) = ymk(0) +

∫ t

0ymk(s)ds. (69)

But ymk(0) → y0 in V and hence in VD, while, for t fixed,∫ t

0 ymk(s)ds ⇀∫ t0 y(s)ds, weakly in VD. So, by taking the weak limit in VD in (69), we

obtain in VD the equality

y(t) = y0 +

∫ t

0y(s)ds, (70)

from which we conclude that y(t) is in VD a.e., with y = y and y(0) = y0.We need now to show that y is actually a solution to the problem (45),

with y(0) = y1. To see this, take a function ' ∈ C1([0, T ]), satisfying'(T ) = 0, and define, for j < m, the function 'j by 'j(t) = '(t)wj , where{wj}m1 was the basis spanning Vm. Now, for a fixed j < m, we multiply (63)with '(t) and integrate to obtain∫ T

0((ym(s), 'j(s))+d(s; ym(s), 'j(s)) + a(s; ym(s), 'j(s)))ds =∫ T

0⟨f(s), 'j(s)⟩V ∗D,VDds.

Noticing that, for each t, we have that d(t; ⋅, 'j(t)) ∈ V ∗D and a(t; ⋅, 'j(t)) ∈V ∗, we find, using the weak convergence above that for m = mk → ∞ andintegration by parts in the first term, that∫ T

0(−(y(s), 'j(s)) + d(s; y(s), 'j(s)) + a(s; y(s), 'j(s)))ds =∫ T

0⟨f(s), 'j(s)⟩V ∗D,VDds+ (y1, 'j(0)), (71)

51

for every j. Now further restrict ' to also satisfy ' ∈ C∞0 (0, T ) and write(71) as ∫ T

0'(s)(−y(s), wj)+∫ T

0'(s)(d(s; y(s), wj) + a(s; y(s), wj)− ⟨f(s), wj⟩V ∗D,VD)ds = 0, (72)

for each j. But by (72), we have

d

dt(y(t), wj) + d(t; y(t), wj) + a(t; y(t), wj) = ⟨f(t), wj⟩V ∗D,VD (73)

for all wj . By of density of ∪∞mVm in V we conclude that y ∈ L2(0, T ;V ∗)and that for all ∈ V

(y(t), ) + d(t; y(t), ) + a(t; y(t), ) = ⟨f(t), ⟩V ∗D,VD , (74)

which was (45). Hence the y we have constructed is indeed a solution tothe equation and by (70) we have that y(0) = y0. In order to verify thaty(0) = y1 we integrate by parts in (71), and by application of (73) we findthat, for all j:

−(y(s), 'j(s))∣s=Ts=0 = (y1, 'j(0)), (75)

or, equivalently(y(0), wj)'(0) = (y1, wj)'(0). (76)

Hence y(0) = y1.In order to prove uniqueness, let y be a solution of our problem (45)

corresponding to (y0, y1, f) = (0, 0, 0), and define for a fixed t1 ∈ (0, T )(arbitrarily chosen) the function by

(t) =

{−∫ t1t y(s)ds for t < t1,

0 for t ≥ t1,(77)

so (T ) = 0. Obviously (t) ∈ V for all t, so we can take (t) = in (45)which yields

⟨y(t), (t)⟩V ∗,V + d(t; y(t), (t)) + a(t; y(t), (t)) = ⟨f(t), (t)⟩V ∗,V . (78)

Because (t) = y(t) for t < t1 (a.e), we have that∫ t1

0(⟨y(t), (t)⟩V ∗,V + ⟨y(t), y(t)⟩V ∗,V )dt =∫ t1

0

d

dt(⟨y(t), (t)⟩V ∗,V )dt = 0, (79)

52

due to (t1) = 0 and the initial conditions. Using this and by integrationof (78) we find∫ t1

0(⟨y(t), y(t)⟩V ∗,V − d(t; y(t), (t))− a(t; y(t), (t)))dt = 0; (80)

hence ∫ t1

0

d

dt(∣y(t)∣2 − a(t; (t), (t)))dt =

2

∫ t1

0(a(t; (t), (t)) + Re d(t; y(t), (t)))dt. (81)

Because (t1) = 0 and y(0) = y0 = 0 this yields

∣y(t1)∣2 + a(0; (0), (0)) = 2

∫ t1

0(a(t; (t), (t)) + Re d(t; y(t), (t)))dt.

(82)From the assumptions on a and a we arrive at

∣y(t1)∣2 + �∣ (0)∣2V ≤ 2

∫ t1

0(c2∣ (t)∣2V + Re d(t; y(t), (t)))dt. (83)

Now notice that

d(t; y(t), (t)) =d

dt(d(t; y(t), (t)))− d(t; y(t), (t))− d(t; y(t), y(t)), (84)

so (from the initial conditions)∫ t1

0d(t; y(t), (t))dt =

∫ t1

0(−d(t; y(t), (t))− d(t; y(t), y(t)))dt. (85)

Because−Re d(t; y(t), y(t)) ≤ �D∣y(t)∣2 − �d∣y(t)∣2VD (86)

we have that

∣y(t1)∣2 + �∣ (0)∣2V ≤

2

∫ t1

0(c2∣ (t)∣2V + �D∣y(t)∣2 − �d∣y(t)∣2VD + Re d(t; y(t), (t)))dt. (87)

Now we introduce the function !(t) =∫ t

0 y(s)ds and use

∣ (t)∣2V = ∣!(t)− !(t1)∣2V ≤ 2∣!(t)∣2V + 2∣!(t1)∣2V (88)

53

to obtain

∣y(t1)∣2 + (�− 4c2t1)∣!(t1)∣2V ≤

2

∫ t1

0(2c2∣!(t)∣2V + �D∣y(t)∣2 − �d∣y(t)∣2VD + Re d(t; y(t), (t)))dt. (89)

Finally we use the differentiability of the damping sesquilinear form d:

∣d(t; y(t), (t))∣ ≤ c4∣y(t)∣VD ∣ (t)∣VD≤ c4

2(�∣y(t)∣2VD +

1

�∣ (t)∣2VD)

≤ c4

2(�∣y(t)∣2VD +

2

�∣!(t)∣2VD +

2

�∣!(t1)∣2VD)

for all � > 0. We now have an inequality

∣y(t1)∣2 + (�− 4c2t1)∣!(t1)∣2V −2c4t1�∣!(t1)∣2VD ≤

2

∫ t1

0(2c2∣!(t)∣2V + �D∣y(t)∣2 − �d∣y(t)∣2VD +

c4

2(�∣y(t)∣2VD +

2

�∣!(t)∣2VD))dt

(90)

and using here that

(�− 4c2t1)∣!(t1)∣2V ≥ (�

2− 2c2t1)(∣!(t1)∣2V + ∣!(t1)∣2VD), (91)

we finally arrive at the formidable inequality

∣y(t1)∣2 + (�

2− 2c2t1)∣!(t1)∣2V + (

�

2− 2c2t1 −

2c4t1�

)∣!(t1)∣2VD ≤

2

∫ t1

0(2c2∣!(t)∣2V + �D∣y(t)∣2 + (

c4�

2− �d)∣y(t)∣2VD +

c4

�∣!(t)∣2VD)dt.

Now fix � > 0 such that ( c4�2 − �d) < 0 and fix t1 = �8(c2+

c4�

)such that

(�2 − 2c2t1− 2c4t1� ) = �

4 . Then also (�2 − 2c2t1) > 0 and the inequality aboveimplies that for some constant M > 0:

∣y(t1)∣2 + ∣!(t1)∣2V + ∣!(t1)∣2VD ≤M∫ t1

0(∣y(t)∣2 + ∣!(t)∣2V + ∣!(t)∣2VD)dt, (92)

and from Gronwall’s inequality we see that y = 0 in the interval [0, t1].Since the length of t1 is independent of the choice of origin, we concludethat y = 0 on [t1, 2t1], etc. Hence y = 0 and uniqueness is proved. That the

54

solution depends continuously on the data is obvious from the inequalitiesused to show existence; indeed, from (67) and (68) and the weak lowersemicontinuity of norms we conclude that the constructed solution satisfies

∣y(t)∣2V + ∣y(t)∣2 + �

∫ t

0∣y(t)∣2VD ≤

K(∣y0∣2V + ∣y1∣2 +

∫ t

0∣f(s)∣2V ∗Dds) (93)

for some positive constants � and K. Integrating from 0 to T yields thedesired result (since (y0, y1, f)→ (y, y) is linear). This completes the proofof Theorem 18.

The proof of Theorem 19 follows from the inequality (92) and the originalproof in the case d = 0 from of [LiMag, p. 275–279], because we do not gainany additional regularity from the form d in this case.

55

17 Inverse or Parameter Estimation Problems

In the generic abstract parameter estimation problem, we consider a dy-namic model of the form (35) where the operators A and D and possiblythe input f depend on some unknown (i.e., to be estimated) functionalparameters q in an admissible family Q ⊂ C1(0, T ;L∞(Ω;Q)) of parame-ters. Here Ω is the underlying set on which the functions of H,V, VD aredefined (e.g., the spatial set Ω = (0, l) in the heat, transport, and beamexamples). We assume that the time dependence of the operators A and Dare through the time dependence of the parameters q(t) ∈ L∞(Ω;Q) whereQ ⊂ Rp is a given constraint set for the values of the parameters. That is,A(t) = A1(q(t)), D(t) = A2(q(t)) so that we have

y(t) +A2(q(t))y(t) +A1(q(t))y(t) = f(t, q) in V ∗

y(0) = y0, y(0) = y1.(94)

Thus we introduce the sesquilinear forms �1, �2 by

a(t;�, ) ≡ �1(q(t))(�, ) = ⟨A1(q(t))�, ⟩V ∗,V , (95)

d(t;�, ) ≡ �2(q(t))(�, ) = ⟨A2(q(t))�, ⟩V ∗D,VD . (96)

It will be convenient in subsequent arguments to use the notation

�i(q)(�, ) ≡ d

dt�i(q)(�, ) (97)

which in the event that �i is linear in q becomes �i(q)(�, ) = �i(q)(�, ).For example, in the Euler Bernoulli example of (52), we would have

�1(q(t))(�, ) =

∫ l

0EI(t, �)�′′(�) ′′(�)d� (98)

�2(q(t))(�, ) =

∫ l

0CDI(t, �)�′′(�) ′′(�)d�, (99)

where q = (EI, CDI) ∈ C1(0, T ;L∞(0, l;R2+)). Note that in this case we do

have �i(q)(�, ) = �i(q)(�, ).In terms of parameter dependent sesquilinear forms we thus will write

(94) as

⟨y(t), �⟩+ �2(q(t))(y(t), �) + �1(q(t))(y(t), �) = ⟨f(t, q), �⟩y(0) = y0, y(0) = y1

(100)

56

for all � ∈ V. As in (36), ⟨⋅, ⋅⟩ denotes the duality product ⟨⋅, ⋅⟩V ∗,V . Insome problems the initial data y0, y1 may also depend on parameters qto be estimated, i.e., y0 = y0(q), y1 = y1(q). We shall not discuss suchproblems here, although the ideas we present can be used to effectively treatsuch problems. We instead refer readers to [13] for discussions of generalestimation problems where not only the initial data but even the underlyingspaces V and H themselves may depend on unknown parameters.

It is assumed that the parameter-dependent sesquilinear forms �1(q), �2(q)of (100) satisfy the continuity and ellipticity conditions (H2)-(H7) of Sec-tion 1 uniformly in q ∈ Q; that is, the constants c1, c2, �, c3, c4, �d, �d of(H2)-(H7) can be found independently of q ∈ Q.

In general inverse problems, one must estimate the functional parametersq from dynamic observations of the system (94) or (100). A fundamentalconsideration in problem formulation involves what will be measured in thedynamic experiments producing the observations. To discuss these mea-surements in a specific setting, we consider the transverse vibrations (forexample, equation (52)) of our beam example where y(t) = y(t, ⋅). Measure-ments, of course, depend on the sensors available. If one considers a trulysmart material structure as in [BSW, RCS], it contains both sensors and ac-tuators which may or may not rely on the same physical device or material.In usual mechanical experiments, there are several popular measurement de-vices [BSW, BT], some of which could possibly be used in a smart materialconfiguration. If one uses an accelerometer placed at the point � ∈ (0, ℓ)along the beam, then one obtains observations y(t, �) of beam accelera-tion. A laser vibrometer will yield data of velocity y(t, �) while proximityprobes including displacement solenoids produce measurements of displace-ment y(t, �). In the case of a beam (or structure) with piezoceramic patches,the patches may be used as sensors as well as actuators. In this case oneobtains observations of voltages which are proportional to the accumulatedstrain; this is discussed fully in [BSW, RCS].

Whatever the measuring devices, the resulting observations can be usedin a maximum likelihood or one of several least squares formulations of theparameter estimation problem, depending on assumptions about the sta-tistical model [7, BT] for errors in the observation process. In the leastsquares formulations, the problems are stated in terms of finding param-eters which give the best fit of the parameter-dependent solutions of thepartial differential equation to dynamic system response data collected aftervarious excitations, while the maximum likelihood estimator results frommaximizing a given (assumed) likelihood function for the parameters, giventhe data.

57

In the beam example, the parameters to be estimated include the stiff-ness coefficient EI(t, �), the Kelvin-Voigt damping parameter cDI(t, �), andany control related parameters that arise in the actuator input f . Detailsregarding the estimation of these parameters in the time independent casefor an experimental beam are given in Section 5.4 of [BSW], while similarexperimental results for a plate are summarized in Section 5.5 of the samereference.

The general ordinary least squares parameter estimation problem can beformulated as follows. For a given discrete set of measured observations z ={zi}Nti=1 corresponding to model observations zob(ti) at times ti as obtainedin most practical cases, we consider the problem of minimizing over q ∈ Qthe least squares output functional

J(q, z) =∣∣∣ C2

{C1{y(ti, ⋅; q)} − {zi}

}∣∣∣2 , (101)

where {y(ti, ⋅; q)} are the parameter dependent solutions of (94) or (100)evaluated at each time ti, i = 1, 2, . . . , Nt and ∣ ⋅ ∣ is an appropriately chosenEuclidean norm. Here the operators C1 and C2 are observation operatorsthat depend on the type of observed or measured data available. The op-erator C1 may have several forms depending on the type of sensors beingused. When the collected data zi consists of time domain displacement, ve-locity, or acceleration values at a point � on the beam as discussed above,the functional takes the form

J�(q, z) =

Nt∑i=1

∣∣∣∣∂�w∂t� (ti, �; q)− zi∣∣∣∣2 , (102)

for � = 0, 1, 2, respectively. In this case the operator C1 involves differenti-ation (either � = 0, 1 or 2 times, respectively) with respect to time followedby pointwise evaluation in t and �. We shall in our presentation of the nextsection adopt the ordinary least squares functional (101) to formulate anddevelop our results.

17.1 Approximation and Convergence

In this section we present a corrected version (Theorem 5.2 of [BSW] containserror in statement and proof) and extension (to treat time dependent coeffi-cients) of arguments for approximation and convergence in inverse problemsfound in Section 5.2 of [BSW]. For more details on general inverse problemmethodology in the context of abstract structural systems, the reader may

58

consult [BSW]. The Banks-Kunisch book [13] contains a general treatmentof inverse problems for partial differential equations in a functional analyticsetting.

The minimization in our general abstract parameter estimation prob-lems for (101) involves an infinite dimensional state space H and an infinitedimensional admissible parameter set Q (of functions). To obtain computa-tionally tractable methods, we thus consider Galerkin type approximationsin the context of the variational formulation (100). Let HN be a sequenceof finite dimensional subspaces of H, and QM be a sequence of finite di-mensional sets approximating the parameter set Q. We denote by PN theorthogonal projections of H onto HN . Then a family of approximating es-timation problems with finite dimensional state spaces and parameter setscan be formulated by seeking q ∈ QM which minimizes

JN (q, z) =∣∣∣ C2

{C1{yN (ti, ⋅; q)} − {zi}

}∣∣∣2 , (103)

where yN (t; q) ∈ HN is the solution to the finite dimensional approximationof (100)) given by

⟨yN (t), �⟩+ �2(q(t))(yN (t), �) + �1(q(t))(yN (t), �) = ⟨f(t, q), �⟩yN (0) = PNy0, yN (0) = PNy1,

(104)

for � ∈ HN . For the parameter sets Q and QM , and state spaces HN , wemake the following hypotheses.

(A1M) The sets Q and QM lie in a metric space Q with metric d. It is assumedthat Q and QM are compact in this metric and there is a mappingiM : Q → QM so that QM = iM (Q). Furthermore, for each q ∈Q, iM (q)→ q in Q with the convergence uniform in q ∈ Q.

(A2N) The finite dimensional subspaces HN satisfy HN ⊂ V as well as theapproximation properties of the next two statements.

(A3N) For each ∈ V, ∣ − PN ∣V → 0 as N →∞.

(A4N) For each ∈ VD, ∣ − PN ∣VD → 0 as N →∞.

The reader is referred to Chapter 4 of [BSW] for a complete discussionmotivating the spaces HN and VD.

We also need some regularity with respect to the parameters q in theparameter dependent sesquilinear forms �1, �2. In addition to (uniform in Q)

59

ellipticity/coercivity and continuity conditions (H2)-(H7), the sesquilinearforms �1 = �1(q), �2 = �2(q) and �1 = �1(q) are assumed to be defined onQ and satisfy the continuity-with-respect-to-parameter conditions

(H8) ∣�1(q)(�, )− �1(q)(�, )∣ ≤ 1d(q, q)∣�∣V ∣ ∣V , for �, ∈ V

(H9) ∣�1(q)(�, )− �1(q)(�, )∣ ≤ 3d(q, q)∣�∣V ∣ ∣V , for �, ∈ V

(H10) ∣�2(q)(�, �)− �2(q)(�, �) ≤ 2d(q, q)∣�∣VD ∣�∣VD , for �, � ∈ VD

for q, q ∈ Q where the constants 1, 2, 3 depend only on Q.Solving the approximate estimation problems involving (103),(104), we

obtain a sequence of parameter estimates {qN,M}. It is of paramount im-portance to establish conditions under which {qN,M} (or some subsequence)converges to a solution for the original infinite dimensional estimation prob-lem involving (100),(101). Toward this goal we have the following results.

Theorem 20 To obtain convergence of at least a subsequence of {qN,M}to a solution q of minimizing (101) subject to (100), it suffices, under as-sumption (A1M), to argue that for arbitrary sequences {qN,M} in QM withqN,M → q in Q, we have

C2C1yN (t; qN,M )→ C2C1y(t; q). (105)

Proof: Under the assumptions (A1M), let {qN,M} be solutions minimiz-ing (103) subject to the finite dimensional system (104) and let qN,M ∈ Q besuch that iM (qN,M ) = qN,M . From the compactness of Q, we may select sub-sequences, again denoted by {qN,M} and {qN,M}, so that qN,M → q ∈ Q andqN,M → q (the latter follows the last statement of (A1M)). The optimalityof {qN,M} guarantees that for every q ∈ Q

JN (qN,M , z) ≤ JN (iM (q), z). (106)

Using (105), the last statement of (A1M) and taking the limit as N,M →∞in the inequality (106), we obtain J(q, z) ≤ J(q, z) for every q ∈ Q, or that qis a solution of the problem for (100),(101). We note that under uniquenessassumptions on the problems (a situation that we hasten to add is not oftenrealized in practice), one can actually guarantee convergence of the entiresequence {qN,M} in place of subsequential convergence to solutions.

We note that the essential aspects in the arguments given above involvecompactness assumptions on the sets QM and Q. Such compactness ideasplay a fundamental role in other theoretical and computational aspects of

60

these problems. For example, one can formulate distinct concepts of problemstability and method stability as in [13] involving some type of continuousdependence of solutions on the observations z, and use conditions similar tothose of (105) and (A1M), with compactness again playing a critical role,to guarantee stability. We illustrate with a simple form of method stability(other stronger forms are also amenable to this approach–see [13]).

We might say that an approximation method, such as that formulatedabove involving QM , HN and (103), is stable if

dist(qN,M (zk), q(z∗))→ 0

as N,M, k →∞ for any zk → z∗ (in this case in the appropriate Euclideanspace), where q(z) denotes the set of all solutions of the problem for (101)and qN,M (z) denotes the set of all solutions of the problem for (103). Here“dist” represents the usual distance set function. Under (105) and (A1M),one can use arguments very similar to those sketched above to establishthat one has this method stability. If the sets QM are not defined througha mapping iM as supposed above, one can still obtain this method stabilityif one replaces the last statement of (A1M) by the assumptions:

(i) If {qM} is any sequence with qM ∈ QM , then there exist q∗ in Q andsubsequence {qMk} with qMk → q∗ in the Q topology.

(ii) For any q ∈ Q, there exists a sequence {qM} with qM ∈ QM such thatqM → q in Q.

Similar ideas may be employed to discuss the question of problem stabilityfor the problem of minimizing (101) over Q (i.e., the original problem) andagain compactness of the admissible parameter set plays a critical role.

Compactness of parameter sets also plays an important role in computa-tional considerations. In certain problems, the formulation outlined above(involving QM = iM (Q)) results in a computational framework wherein theQM and Q all lie in some uniform set possessing compactness properties.The compactness criteria can then be reduced to uniform constraints onthe derivatives of the admissible parameter functions. There are numericalexamples (for example, see [BI86]) which demonstrate that imposition ofthese constraints is necessary (and sufficient) for convergence of the result-ing algorithms. (This offers a possible explanation for some of the numericalfailures [YY] of such methods reported in the engineering literature.)

Thus we have that compactness of admissible parameter sets play a fun-damental role in a number of aspects, both theoretical and computational,

61

in parameter estimation problems. This compactness may be assumed (andimposed) explicitly as we have outlined here, or it may be included implicitlyin the problem formulation through Tikhonov regularization as discussed forexample by Kravaris and Seinfeld [KS], Vogel [Vog] and widely by manyothers. In the regularization approach one restricts consideration to a sub-set Q1 of parameters which has compact embedding in Q and modifies theleast-squares criterion to include a term which insures that minimizing se-quences will be Q1 bounded and hence compact in the original parameterset Q.

After this short digression on general inverse problem concepts, we returnto the condition (105). To demonstrate that this condition can be readilyestablished in many problems of interest to us here, we give the followinggeneral convergence results.

Theorem 21 Suppose that HN satisfies (A2N),(A3N),(A4N) and assumethat the sesquilinear forms �1(q), �1(q) and �2(q) satisfy (H8),(H9),(H10),respectively, as well as (H1)-(H7) of Section 1 (uniformly in q ∈ Q). Fur-thermore, assume that

q → f(⋅; q) is continuous from Q to L2(0, T ;V ∗D). (107)

Let qN be arbitrary in Q such that qN → q in Q. Then if in addition y ∈L2(0, T ;V ), we have as N →∞,

yN (t; qN )→ y(t; q) in V norm for each t > 0

yN (t; qN )→ y(t; q) in L2(0, T ;VD) ∩ C(0, T ;H),

where (yN , yN ) are the solutions to (104) and (y, y) are the solutions to(100).

Proof: From Theorem 19 of Section 1 we find that the solution of (100)satisfies y(t) ∈ V for each t, y(t) ∈ VD for almost every t > 0. Because

∣yN (t; qN )− y(t; q)∣V ≤ ∣yN (t; qN )− PNy(t; q)∣V + ∣PNy(t; q)− y(t; q)∣V ,

and (A3N) implies that second term on the right side converges to 0 asN →∞, it suffices for the first convergence statement to show that

∣yN (t; qN )− PNy(t; q)∣V → 0 as N →∞.

Similarly, we note that this same inequality with yN , y replaced by yN , yand the V -norm replaced by the VD-norm along with (A4N) permits us toclaim that the convergence

∣yN (t; qN )− PN y(t; q)∣VD → 0 as N →∞,

62

is sufficient to establish the second convergence statement of the theorem.We shall, in fact, establish the convergence of yN − PN y in the stronger Vnorm.

Let yN = yN (t; qN ), y = y(t; q), and ΔN = ΔN (t) ≡ yN (t; qN ) −PNy(t; q). Then

ΔN = yN − d

dtPNy = yN − PN y

and

ΔN = yN − d2

dt2PNy

because y ∈ L2((0, T ), VD), y ∈ L2((0, T ), V ∗). We suppress the dependenceon t in the arguments below when no confusion will result. From (100) and(104), we have for ∈ HN

⟨ΔN , ⟩V ∗,V = ⟨yN − y + y − d2

dt2PNy, ⟩V ∗,V

= ⟨f(qN ), ⟩V ∗D,VD − �2(qN )(yN , )− �1(qN )(yN , )

− ⟨f(q), ⟩V ∗D,VD + �2(q)(y, ) + �1(q)(y, )

+ ⟨y − d2

dt2PNy, ⟩V ∗,V .

This can be written as

⟨ΔN , ⟩V ∗,V + �1(qN )(ΔN , )

= ⟨y − d2

dt2PNy, ⟩V ∗,V − ⟨f(q)− f(qN ), ⟩V ∗D,VD

+ �2(qN )(y − PN y, ) + �2(q)(y, )− �2(qN )(y, )

+ �1(qN )(y − PNy, ) + �1(q)(y, )− �1(qN )(y, )

− �2(qN )(ΔN , ).

(108)

Choosing ΔN as the test function in (108) and employing the equality⟨ΔN , ΔN ⟩V ∗,V = 1

2ddt ∣Δ

N ∣2H (this follows using definitions of the dualitymapping - see [?] and the hypothesis (A2N)), we have using the symmetry

63

of �1

1

2

d

dt

{∣ΔN ∣2H+�1(qN )(ΔN ,ΔN )

}=Re

{⟨y − d2

dt2PNy, ΔN ⟩V ∗,V − ⟨f(q)− f(qN ), ΔN ⟩V ∗D,VD

+ �2(qN )(y − PN y, ΔN ) + �2(q)(y, ΔN )

− �2(qN )(y, ΔN ) + �1(qN )(y − PNy, ΔN )

+ �1(q)(y, ΔN )− �1(qN )(y, ΔN )− �2(qN )(ΔN , ΔN )

+ �1(qN )(ΔN ,ΔN )}.

(109)

We observe that ⟨y − d2

dt2PNy, ΔN ⟩V ∗,V ≡ 0 because PN is an orthogonal

projection. Thus, we find

1

2

d

dt

{∣ΔN ∣2H + �1(qN )(ΔN ,ΔN )

}= Re

{−TN3 + �2(qN )(y − PN y, ΔN ) + TN2 +

d

dt(�1(qN )(y − PNy,ΔN ))

− �1(qN )(d

dt(y − PNy),ΔN )− �1(qN )(y − PNy,ΔN )

+ TN1 + �2(qN )(ΔN , ΔN ) + �1(qN )(ΔN ,ΔN )},

(110)

where

TN1 = Δ�N1 (y, ΔN ) ≡ �1(q)(y, ΔN )− �1(qN )(y, ΔN )

TN2 = Δ�N2 (y, ΔN ) ≡ �2(q)(y, ΔN )− �2(qN )(y, ΔN )

TN3 = ⟨ΔfN , ΔN ⟩V ∗D,VD ≡ ⟨f(q)− f(qN ), ΔN ⟩V ∗D,VD .(111)

Note that here we have used that y ∈ L2(0, T ;V ). Integrating the terms in(110) from 0 to t and using the initial conditions

ΔN (0) = yN (0)− PNy(0) = yN (0)− PNy0 = 0

ΔN (0) = yN (0)− PN y(0) = yN (0)− PNy1 = 0,

we obtain (here we do include arguments (t) and (s) in our calculations and

64

estimates to avoid confusion)

1

2∣ΔN (t)∣2H + �1(qN (t))(ΔN (t),ΔN (t))

=

∫ t

0

{Re{�2(qN (s))(y(s)− PN y(s), ΔN (s))

− �1(qN (s))(y(s)− PN y(s),ΔN (s))− �1(qN (s))(y(s)− PNy(s),ΔN (s))

− �2(qN (s))(ΔN (s), ΔN (s)) + TN1 (s)

+ TN2 (s) + TN3 (s) + �1(qN (s))(ΔN (s),ΔN (s))}}ds

+Re{�1(q(t))(y(t)− PNy(t),ΔN (t))

}. (112)

We consider the TNi terms in this equation. We have

TN1 =d

dtΔ�N1 (y,ΔN )−Δ�N1 (y,ΔN )−Δ�N1 (y,ΔN )

so that∫ t

0TN1 (s)ds = Δ�N1 (y(t),ΔN (t))

−∫ t

0

{Δ�N1 (y(s),ΔN (s))−Δ�N1 (y(s),ΔN (s))

}ds.

Using (H8),(H9) we thus obtain

Re

∫ t

0TN1 (s)ds ≤ 2

1

4�d(qN , q)2∣y(t)∣2V + �∣ΔN (t)∣2V

+

∫ t

0

{ 21

4�d(qN , q)2∣y(s)∣2V + �∣ΔN (s)∣2V

+ 2

3

4�d(qN , q)2∣y(s)∣2V + �∣ΔN (s)∣2V

}ds. (113)

Similarly, using (H10) we find

Re

∫ t

0(TN2 (s) + TN3 (s))ds ≤∫ t

0

{ 22

4�d(qN , q)2∣y(s)∣2V + 2�∣ΔN (s)∣2VD +

1

4�∣ΔfN (s)∣2V

}ds. (114)

65

These can then be used in (112) to obtain the estimate

1

2∣ΔN (t)∣2H+�∣ΔN (t)∣2V +

∫ t

0�d∣ΔN (s)∣2VDds

≤∫ t

0

{�d∣ΔN (s)∣2H +

c23

4�∣y(s)− PN y(s)∣2VD + 3�∣ΔN (s)∣2VD

+c2

1

2∣y(s)− PN y(s)∣2V +

c22

2∣y(s)− PNy(s)∣2V + (1 + �+ c2)∣ΔN (s)∣2V

+ 2

1

4�d(qN , q)2∣y(s)∣2V +

23

4�d(qN , q)2∣y(s)∣2V +

22

4�d(qN , q)2∣y(s)∣2V

+1

4�∣ΔfN (s)∣2V

}ds+

c21

4�∣y(t)− PNy(t)∣2V + 2�∣ΔN (t)∣2V +

21

4�d(qN , q)2∣y(t)∣2V .

This finally reduces to

1

2∣ΔN (t)∣2H + (�− 2�)∣ΔN (t)∣2V +

∫ t

0(�d − 3�)∣ΔN (s)∣2VDds

≤∫ t

0

{�d∣ΔN (s)∣2H + (1 + �+ c2)∣ΔN (s)∣2V

}ds

+c2

1

4�∣y(t)− PNy(t)∣2V +

∫ t

0

{c23

4�∣y(s)− PN y(s)∣2VD +

c21

2∣y(s)− PN y(s)∣2V

}ds︸ ︷︷ ︸

�N1 (t)

+ 2

1

4�d(qN , q)2∣y(t)∣2V +

∫ t

0

{ 1

4�∣ΔfN (s)∣2V +

c22

2∣y(s)− PNy(s)∣2V

}ds︸ ︷︷ ︸

�N2 (t)

+

∫ t

0

{ 21

4�d(qN , q)2∣y(s)∣2V +

23

4�d(qN , q)2∣y(s)∣2V +

22

4�d(qN , q)2∣y(s)∣2V +

}ds︸ ︷︷ ︸

�N3 (t)

.

Therefore, under the assumptions we have as N →∞ and qN → q

supt∈(0,T )

[�N1 (t) + �N2 (t) + �N3 (t)]→ 0.

Applying Gronwall’s inequality, we find that

ΔN → 0 in C(0, T ;H)

ΔN → 0 in C(0, T ;V )

ΔN → 0 in L2(0, T ;VD),

66

and hence the convergence statement of the theorem holds.Remark: The condition “y ∈ L2(0, T ;V )” routinely occurs if the struc-

ture under investigation has sufficiently strong damping so that VD is equiv-alent to V .

17.2 Some Further Remarks

We have provided sufficient conditions and detailed arguments for existenceand uniqueness of solutions to abstract second order non-autonomous hy-perbolic systems such as those with time dependent “stiffness”, “damping”and input parameters. In addition, we have considered a class of correspond-ing inverse problems for these systems and argued convergence results forapproximating problems that yield a type of method stability as well as aframework for finite element type computational techniques. The efficacyof such methods have been demonstrated for autonomous systems in earlierefforts [BSW]. The approaches developed here can readily be extended (al-beit with considerable tedium) for higher dimensional spatial systems suchas plates, shells, etc.

67

18 “Weak” or “Variational Form”

We consider the origin of the terms “weak or variational form” as opposedto strong or closed form of PDE’s. We use the beam equation to illustrateideas.

Recall Example 6, the cantilever beam. This example, given in classicalform (which can be derived in a straight forward manner using force andmoment balance–see [BT]) is

�∂2y

∂t2+

∂y

∂t+

∂2

∂�2

(EI

∂2y

∂�2+ cDI

∂3y

∂�2∂t

)= f(t, �)

with boundary conditions

y(t, 0) = 0∂y∂� (t, 0) = 0

(115)

(EI ∂

2y∂�2

+ cDI∂3y∂�2∂t

)∣�=l = 0

∂∂�

[(EI ∂

2y∂�2

+ cDI∂3y∂�2∂t

)]∣�=l = 0

(116)

and initial conditionsy(0, �) = Φ(�)∂y∂t (0, �) = Ψ(�)

To facilitate our discussions, we consider an undamped and unforcedversion (i.e., = cDI = 0, f = 0) of the above system. Rather than forceand moment balance, we consider energy formulations for the beam. For asegment of the beam in [�, � + Δ�], one can argue that the kinetic energy(at a given time t) is given by

KE(t) = T (t) =1

2

∫ �+Δ�

��(∂y

∂t(t, �))2d�,

and hence the kinetic energy of the entire beam is given by

T (t) =1

2

∫ l

0�y2(t, �)d�.

Similarly, the potential (or strain) energy U of the beam at any given timet is given by

PE(t) = U(t) =1

2

∫ l

0EI(

∂2y

∂�2)2d�.

68

A fundamental tenant of the mechanics of rigid or elastic bodies is Hamil-ton’s “Principle of Stationary Action” (often, in a misnomer, referred to asHamilton’s principle of “least action”) which postulates that any systemundergoing motion during a period [t0, t1] will exhibit motion y(t, �) thatprovides the “least action” for the system with a stationary value. The“Action” is defined by

A =∫ t1t0

(KE − PE)dt

=∫ t1t0

[T (t)− U(t)]dt.

For the beam of Example 6, this means that the motion or vibrations y(t, �)must provide a stationary value to the action

A[y] =

∫ t1

t0

∫ l

0[1

2�y2 − 1

2EI(y′′)2]d�dt

integrated over any time interval [t0, t1]. Through the calculus of variations(a field of mathematics that was the precursor to modern control theory),this leads to an equation of motion for the vibrations y that the beam motionmust satisfy.

To further explore this, we consider y(t, �) as the motion of the beamand consider a family of variations y(t, �)+ ��(t, �) where � is chosen so thaty + �� is an “admissible variation”, i.e., y + �� must satisfy the essentialboundary conditions (115).

We define V = H2L(0, l) = {' ∈ H2(0, l)∣'(0) = '′(0) = 0}. Let ∈

C2(t0, t1) with (t0) = (t1) = 0. Then � ∈ N ≡ {�∣� = ',' ∈ V }satisfies � is C2 in t, H2 in � with �(t0, �) = �(t1, �) = 0 and �(t, 0) =�′(t, 0) = 0. Then by Hamilton’s principle, we must have that A[y + ��] for� > 0, � ∈ N , must have a stationary value at � = 0. That is,

d

d�A[y + ��]∣�=0 = 0. (117)

Since

A[y + ��] =

∫ t1

t0

∫ l

0[1

2�(y + ��)2 − 1

2EI(y′′ + ��′′)2]d�dt,

we find from (117) the weak (in time t and space �) form of the equationgiven by

0 =

∫ t1

t0

∫ l

0[�y� − EIy′′�′′]d�dt (118)

69

for all � ∈ N .To explore further, we formally integrate by parts in the first term (with

respect to t) to obtain∫ t1t0

∫ l0 �y�d�dt = −

∫ t1t0

∫ l0 �y�d�dt+

∫ l0 �y�d�∣

t=t1t=t0

= −∫ t1t0

∫ l0 �y�d�dt

since �(t0, �) = �(t1, �) = 0. Since � has the form � = ', equation (118)has the form ∫ t1

t0

∫ l

0[�y'+ EIy′′'′′] d�dt = 0 (119)

for all ∈ C2[t0, t1] with (t0) = (t1) = 0, and all ' ∈ V . Since this holdsfor arbitrary , we must have in the L2(t0, t1) sense∫ l

0[�y'+ EIy′′'′′]d� = 0 for all ' ∈ V.

In our former notation of Gelfand triples with V = H2L(0, l) and H =

L2(0, l), this may be written

⟨�y, '⟩V ∗,V + ⟨EIy′′, '′′⟩H = 0 for all ' ∈ V

in the L2(t0, t1) sense, which is exactly the “weak” or “variational” form ofthe beam equation we have encountered previously. Note that in fact thetrue variational form was given in (118); that is,∫ t1

t0

[−⟨�y, '⟩ + ⟨EIy′′, '′′⟩ ]dt = 0

for all ' ∈ V and ∈ C2[t0, t1] with (t0) = (t1) = 0. (See the proofs andour remarks concerning solutions in the L2(t0, t1;V )∗ ∼= L2(t0, t1;V ∗) sensein the well-posedness (existence) results for second order systems discussedearlier in Section 16.

We note that if the variational solution y has additional smoothnessso that y ∈ V

∩H4(0, l) (more precisely EIy′′ ∈ H2(0, l)), then we can

integrate by parts twice (with respect to �) in the second term of (119) toobtain in place of (119):∫ t1

t0

∫ l

0[�y'+ (EIy′′)′′'] d�dt+

∫ t1

t0

−(EIy′′)'′∣�=l�=0 dt

+

∫ t1

t0

(EIy′′)′'∣�=l�=0 dt = 0

70

for ' ∈ V, ∈ C2[t0, t1] with (t0) = (t1) = 0. This can be written∫ t1

t0

[∫ l

0[�y'+ (EIy′′)′′']d� +−EIy′′'′∣�=l + (EIy′′)′'∣�=l

] dt = 0

for arbitrary ' ∈ V . We note once again that this results in the strong orclassical form of the equations

�∂2y

∂t2+

∂2

∂�2(EI

∂2y

∂�2) = 0

with the essential boundary conditions

y(t, 0) = y′(t, 0) = 0

as well as the natural boundary conditions

EI∂2y

∂�2(t, l) =

∂

∂�(EI

∂2y

∂�2)(t, l) = 0

holding.

We remark on an additional perspective of the weak or variational orenergy formulation of dynamical systems such as the beam equation. Theweak or variational form may be thought of as Euler’s equations in thecalculus of variations. If

J(y) =

∫F (t, y, y)dt = A[y],

then the condition of stationarity

d

d�J(y + ��)∣�=0 = 0

implies ∫(Fy� + Fy�)dt = 0

which is the “true” Euler’s equation. We may then invoke the duBois Rey-mond lemma below to obtain

∂F

dy=

d

dt

∂F

dy

in a distributional sense. Specifically, we have

71

Lemma 1 (duBois Raymond’s Lemma)If ∫

(G1� +G2�) = 0

for all �, thend

dtG2 = G1

in a weak sense. In other words, G1 is the distributional derivative of G2.

If we assume enough smoothness and integrate by parts, we obtain thestrong form of Euler’s equation:

− d

dtFy(t, y, y) + Fy(t, y, y) = 0.

In the derivation above, we considered the undamped and unforced ver-sion of the equation. In the case of the forced beam, we can add a conser-vative force term W = fy in our derivation, and we will obtain the desired⟨f, '⟩ term in our result. However, there is no known way to derive the weakform with damping. In other words, Hamilton’s principle is essentially validfor conservative forces, but it doesn’t conveniently handle nonconservative(dissipative) forces (damping).

72

19 Finite Element Approximations

We will now consider finite element approximations or Galerkin approxima-tions for general first order systems for which parabolic systems are specialcases. Consider {

x(t) = Ax+ F in V ∗ (H if possible)x(0) = x0

(120)

where V ↪→ H ↪→ V ∗ is the usual Gelfand triple. We can write the abovesystem in the weak or variational form (i.e., in V ∗ ) as{

⟨x(t), '⟩V ∗,V + �(x(t), ') = ⟨F (t), '⟩V ∗,Vx(0) = x0

for ' ∈ V . If � is V continuous and V -elliptic, and S(t) ∼ eAt (i.e., A is theinfinitesimal generator of a C0 semigroup S(t)), we can write

x(t) = S(t)x0 +

∫ t

0S(t− �)F (�)d� (121)

where x ∈ L2(0, T ;V )∩C(0, T ;H) and x ∈ L2(0, T ;V ∗). We can use this

formulation to give a nice treatment of finite element approximations ofGalerkin type.

In general, this is an infinite dimensional space; therefore, we want toproject the system into a finite dimensional space in which we can compute.Let HN = span{BN

1 , BN2 , ..., B

NN } ⊂ V be the approximation of H. Typical

choices for the BNj are piecewise linear or piecewise cubic splines (e.g., see

[13]). The idea is to replace (120) by{xN (t) = ANxN (t) + FN (t) in HN

xN (0) = xN0 ,

or equivalently, replace (121) by

xN (t) = SN (t)xN0 +

∫ t

0SN (t− �)FN (�)d�

where SN (t) ∼ eAN t.One of the key constructs we need is PN : H → HN which is called the

orthogonal projection of H onto HN . In other words, PN is defined by

⟨PN'− ', ⟩ = 0 for all ∈ HN

73

or∣PN'− '∣H = inf

∈HN∣ − '∣H .

We would like FN → F and xN0 → x0, so we take xN0 = PNx0 and FN (t) =PNF (t). We also want AN ∈ ℒ(HN ) and AN ≈ A. However, we have

defined SN (t) = eAN t and S(t) ∼ eAt; therefore, if we had SN (t) → S(t),

then we would be able to argue xN (t) → x(t). This is considered in theTrotter-Kato theorem which will be discussed later.

Now to relate this to the computational aspects of finite elements, wefirst restrict the equations

⟨x(t), '⟩+ �1(x(t), ') = ⟨F (t), '⟩ for all ' ∈ V (122)

to HN ×HN . In other words, let

xN (t) =

N∑j=1

wNj (t)BNj

be a trial solution with

xN (0) =

N∑j=1

wN0jBNj .

Substituting this into (122), we have

⟨N∑j=1

wNj (t)BNj , '⟩+ �1(

N∑j=1

wNj (t)BNj , ') = ⟨F (t), '⟩ (123)

for ' ∈ HN . Successively choose ' = BN1 , B

N2 , ...B

NN in (123). From this we

obtain an N ×N vector system for wN (t) = (wN1 (y), .., wNN (t))T given by

N∑j=1

wNj (t)⟨BNj , B

Ni ⟩+

N∑j=1

wNj (t)�(BNj , B

Ni ) = ⟨F (t), BN

i ⟩ (124)

for i = 1, 2, ..., N .Using standard engineering and applied mathematics terminology, we

define the mass matrix MN = (⟨BNi , B

Nj ⟩), the stiffness matrix KN =

(�(BNi , B

Nj )), and the column vector FN (t) = (⟨F (t), BN

i ⟩). Then (124)can be written {

MN wN (t) +KNwN (t) = FN (t)wN (0) = wN0

(125)

74

or {wN (t) = −(MN )−1KNwN (t) + (MN )−1FN (t)wN (0) = wN0 .

Considering wN0 , we have xN (0) = PNx0 which implies ⟨PNx0−x0, BNi ⟩ =

0 for i = 1, ..., N . However, xN0 =∑N

j=1wN0jB

Nj . Therefore,

⟨N∑j=1

wN0jBNj − x0, B

Ni ⟩ = 0

for i = 1, ..., N which gives

N∑j=1

wN0j⟨BNj , B

Ni ⟩ = ⟨x0, B

Ni ⟩.

Defining wN0 = col(wN01, ..., wN0N ), we have

wN0 = (MN )−1col(⟨x0, BNi ⟩).

From this, our system for w becomes{wN (t) = −(MN )−1KNwN (t) + (MN )−1FN (t)wN0 = (MN )−1col(⟨x0, B

Ni ⟩).

However, we normally do not solve the system in this form. If ⟨Bi, Bj⟩ = 0for i ∕= j, then MN is diagonal and the system of the form (125) is an easiersystem with which to work. More generally, the (finite element) system issolved in the form

MN wN (t) = −KNwN (t) + FN (t)

MNwN0 = col(⟨x0, BNi ⟩),

rather than inverting the matrix MN which is, if not diagonal, usually abanded matrix (tri-banded for piecewise linear splines, seven banded forcubic splines, etc, –see [13]), lending itself to fast algebraic solvers (e.g.,those based on LU decompositions).

75

20 Trotter-Kato Approximation Theorem

The Trotter-Kato Approximation Theorem is the functional analysis versionof the Lax Equivalence Principle used in finite difference approximation forPDE’s which dates back to the 1960’s. The fundamental ideas underlying theLax Equivalence Principle is that “consistency” and “stability” are achievedif and only if we have “convergence” of our system. If we have a PDE

utt = Au

and an approximationuNtt = ANuN

then consistency refers to AN → A in some sense. Stability refers to ∣eAN t∣ ≤Me!t, and convergence means eA

N t → eAt in some sense. For relevant andmuch more detailed discussions, see [RM].

There are two different versions of the Trotter-Kato theorem which wewill discuss here. We will first consider the operator convergence form of theTrotter-Kato theorem.

Theorem 22 Let X and XN be Hilbert spaces such that XN ⊂ X. LetPN : X → XN be an orthogonal projection of X onto XN . Assume PNx→x for all x ∈ X. Let AN , A be infinitesimal generators of C0 semigroupsSN (t), S(t) on XN , X respectively satisfying

(i) there exists M,! such that ∣SN (t)∣ ≤Me!t for each N

(ii) there exists D dense in X such that for some �, (�I − A)D is densein X and ANPNx→ Ax for all x ∈ D.

Then for each x ∈ X, SN (t)PNx→ S(t)x uniformly in t on compact inter-vals [0, T ].

The arguments for this result are given in [Pa, Chapter 3, Theorem 4.5 ].We next will examine theresolvent convergence version of the Trotter-Katotheorem. This form is a modification of the previous version.

Theorem 23 Replace (ii) in the above theorem by (ii) defined as

(ii) There exists � ∈ �(A)∞∩N=1

�(AN ) with Re(�) > ! so that R�(AN )PNx

→ R�(A)x for each x ∈ X.

76

Then under this and the remaining conditions in the above theorem, theconclusions also hold.

For proofs, see [Pa, Chap. 3, Theorems 4.2,4.3,4.4]. See also [13, Theo-rem 1.14]. Convergence rates are given in [13, Theorem 1.16].

For certain problems, it is not necessary for HN ⊂ dom(A) which carriesboth the essential and natural boundary conditions. We may need to onlychoose an appropriate approximation HN such that HN ⊂ V which carriesjust the essential boundary conditions. If we restrict ourselves to first ordersystems in the context of the Gelfand triple V ↪→ H ↪→ V ∗ with HN ap-proximating V as N → ∞ then we have a special case of the Trotter-Katotheorem.

Let the condition (C1) be denoted by(C1) For each z ∈ V , there exists zN ∈ HN such that ∣z − zN ∣V → 0 asN →∞.

Suppose � is V -elliptic, i.e., Re�(',') ≥ �∣'∣2V for � > 0. Also assume� is V continuous, i.e., ∣�(', )∣ ≤ ∣'∣V ∣ ∣V . Let PN : H → HN be anorthogonal projection. Then

∣PNz − z∣ = inf{∣zN − z∣H ∣ zN ∈ HN}.

Under (C1), we have ∣PNz − z∣H ≤ ∣zN − z∣H ≤ ∣zN − z∣V → 0 as N →∞.Therefore, under (C1), PNz → z for z ∈ H.

Next we define AN . We have �(', ) = ⟨−A', ⟩H for ' ∈ dom(A) ={ ∈ V ∣A ∈ H}, ∈ V where A is an infinitesimal generator of a C0

semigroup of contractions on H, i.e., ∣eAt∣ ≤ 1. We define AN through therestriction of � to HN ×HN . Therefore, AN : HN → HN is defined by

�('N , N ) = ⟨−AN'N , N ⟩H 'N , N ∈ HN

= ⟨−A'N , N ⟩V ∗,V .

By V -ellipticity, we obtain AN is an infinitesimal generator of a contractionsemigroup on HN , i.e., ∣eAN t∣ ≤ 1.

Theorem 24 If � is V -elliptic, V continuous and (C1) holds, then

R�(AN )PNz → R�(A)z

in the V norm for z ∈ H and � = 0.

See [BI] for additional discussion.

77

Proof

Let z ∈ H and take � = 0. Now define wN = R�(AN )PNz and w = R�(A)zwhere �(', ) = ⟨−A', ⟩H , ' ∈ dom(A), ∈ V . By definition, we havew ∈ dom(A).

By (C1), there exists wN ∈ HN such that ∣wN − w∣V → 0 as N → ∞.Let zN = wN − wN . We need to show zN → 0 in V .

Since R�(A) = (�I −A)−1, we have

�(w, zN ) = ⟨−AR�(A)z, zN ⟩H= ⟨z, zN ⟩

while�(wN , zN ) = ⟨−ANR�(AN )z, zN ⟩H

= ⟨z, zN ⟩,and hence these are equal. Thus

�∣zN ∣2V ≤ �(zN , zN )= �(wN , zN )− �(wN , zN )= �(w, zN )− �(wN , zN )= �(w − wN , zN )≤ ∣w − wN ∣V ∣zN ∣V .

Therefore, we have �∣zN ∣V ≤ ∣w− wN ∣V . However, ∣wN −w∣V → 0 implies∣zN ∣V → 0 and thus ∣w − wN ∣V ≤ ∣w − wN ∣V + ∣wN − wN ∣V → 0.

Remark: Theorem 2.2 of [BI] is a parameter dependent version of this,i.e., A = A(q), AN = AN (qN ), q, qN ∈ Q, which can be used in inverseproblems.

Theorem 25 Suppose � is V -elliptic, V continuous, and (C1) holds. ThenSN (t)PNz → S(t)z in the V norm for each z ∈ H uniformly in t on compactintervals.

Proof

The arguments involve using Tanabe type estimates after considering anappropriate setting and using the Trotter-Kato, first in H and then in V .Let X = H,XN = HN , PN : H → HN be an orthogonal projection. ThenPNz → z for all z ∈ H by (C1). Convergence in H norm follows immediatelyfrom application of the Trotter-Kato in H. To obtain V convergence issomewhat more work and more delicate; for these details we refer the readerto [BI, Theorem 2.3].

78

References

[A] R. A. Adams. Sobolev Spaces. Academic Press, 1975.

[Aubin] J.-P. Aubin, Optima and Equilbria: An Introduction to NonlinearAnalysis, Spring-Verlag, Berlin Heidelberg, 1993.

[BJR] A. Bamberger, P. Joly and J. E. Roberts, Second-order absorbingboundary conditions for the wave equation: a solution for the cornerproblem, SIAM J. Numer. Anal., 27 (1990), 323–352.

[BBDS1] H.T. Banks, J.E. Banks, L.K. Dick, and J.D. Stark, Estimationof dynamic rate parameters in insect populations undergoing sublethalexposure to pesticides, CRSC-TR05-22, NCSU, May, 2005; Bulletin ofMathematical Biology, 69 (2007), 2139–2180.

[BBDS2] H.T. Banks, J.E. Banks, L.K. Dick, and J.D. Stark, Time-varyingvital rates in ecotoxicology: selective pesticides and aphid populationdynamics, Ecological Modelling, 210 (2008), 155–160.

[BBJ] H. T. Banks, J. E. Banks and S. L. Joyner, Estimation in time-delay modeling of insecticide-induced mortality, CRSC-TR08-15, Octo-ber, 2008; J. Inverse and Ill-posed Problems, 17 (2009), 101–125.

[BBi] H. T. Banks and K. L. Bihari, Modeling and estimating uncertaintyin parameter estimation, Inverse Problems, 17 (2001) 1–17.

[BBo] H.T. Banks and V.A. Bokil, Parameter identification for dispersivedielectrics using pulsed microwave interrogating signals and acousticwave induced reflections in two and three dimensions, CRSC-TR04-27,July, 2004; Revised version appeared as A computational and statisticalframework for multidimensional domain acoustoptic material interroga-tion, Quart. Applied Math., 63 (2005), 156–200.

[Shrimp] H.T.Banks, V.A. Bokil, S. Hu, F.C.T. Allnutt, R. Bullis, A.K.Dhar and C.L. Browdy, Shrimp biomass and viral infection for produc-tion of biological countermeasures, CRSC-TR05-45, NCSU, December,2005; Mathematical Biosciences and Engineering, 3 (2006), 635–660.

[BBH] H. T. Banks, D. M. Bortz and S. E. Holte, Incorporation of vari-ability into the mathematical modeling of viral delays in HIV infectiondynamics, Math. Biosciences, 183 (2003), 63–91.

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[BBPP] H.T. Banks, D.M. Bortz, G.A. Pinter and L.K. Potter, Modelingand imaging techniques with potential for application in bioterrorism,CRSC-TR03-02, January, 2003; Chapter 6 in Bioterrorism: Mathe-matical Modeling Applications in Homeland Security, (H.T. Banks andC. Castillo-Chavez, eds.), Frontiers in Applied Math, FR28, SIAM,Philadelphia, PA, 2003, 129–154.

[BBKW] H.T. Banks, L.W. Botsford, F. Kappel, and C. Wang, Modelingand estimation in size structured population models, LCDS-CCS Re-port 87-13, Brown University; Proceedings 2nd Course on MathematicalEcology, (Trieste, December 8-12, 1986) World Press (1988), Singapore,521–541.

[BBL] H.T. Banks, M.W. Buksas, T. Lin. Electromagnetic Material Inter-rogation Using Conductive Interfaces and Acoustic Wavefronts. SIAMFR 21, Philadelphia, 2002.

[BBu1] H.T. Banks and J.A. Burns, An abstract framework for approx-imate solutions to optimal control problems governed by hereditarysystems, International Conference on Differential Equations (H. An-tosiewicz, Ed.), Academic Press (1975), 10–25.

[BBu2] H.T. Banks and J.A. Burns, Hereditary control problems: numericalmethods based on averaging approximations, SIAM J. Control & Opt.,16 (1978), 169–208.

[BDSS] H.T. Banks, M. Davidian, J.R. Samuels Jr., and K. L. Sutton, An in-verse problem statistical methodology summary, Center for Research inScientific Computation Report, CRSC-TR08-01, January, 2008, NCSU,Raleigh; Chapter XX in Statistical Estimation Approaches in Epidemi-ology, (edited by Gerardo Chowell, Mac Hyman, Nick Hengartner, LuisM.A Bettencourt and Carlos Castillo-Chavez), Springer, Berlin Heidel-berg New York, to appear.

[BD] H.T. Banks and J.L. Davis, A comparison of approximation methodsfor the estimation of probability distributions on parameters, CRSC-TR05-38, NCSU, October, 2005; Applied Numerical Mathematics, 57(2007), 753–777.

[BDTR] H.T. Banks and J.L. Davis, Quantifying uncertainty in the estima-tion of probability distributions with confidence bands, CRSC-TR07-21, NCSU, December, 2007; Mathematical Biosciences and Engineer-ing, 5 (2008), 647–667.

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[GRD-FP] H.T. Banks, J.L. Davis, S.L. Ernstberger, S. Hu, E. Artimovich,A.K. Dhar and C.L. Browdy, A comparison of probabilistic and stochas-tic formulations in modeling growth uncertainty and variability, CRSC-TR08-03, NCSU, February, 2008; Journal of Biological Dynamics, 3(2009) 130–148.

[BDEHAD] H.T. Banks, J.L. Davis, S.L. Ernstberger, S. Hu, E. Artimovichand A.K. Dhar, Experimental design and estimation of growth ratedistributions in size-structured shrimp populations, CRSC-TR08-20,NCSU, November, 2008; Inverse Problems, to appear.

[GRD-FP2] H.T. Banks, J.L. Davis, and S. Hu, A computational compar-ison of alternatives to including uncertainty in structured populationmodels, CRSC-TR09-14, June, 2009; in Three Decades of Progress inSystems and Control, Springer, 2009, to appear.

[BDEK] H.T. Banks, S. Dediu, S. L. Ernstberger and F. Kappel, Gener-alized sensitivities and optimal experimental design, CRSC-TR08-12,September, 2008, Revised Nov, 2009; J Inverse and Ill-Posed Problems,submitted.

[BF] H.T. Banks and B.G. Fitzpatrick, Estimation of growth rate distri-butions in size-structured population models, CAMS Tech. Rep. 90-2,University of Southern California, January, 1990; Quarterly of AppliedMathematics, 49 (1991), 215–235.

[BFPZ] H.T. Banks, B.G. Fitzpatrick, L.K. Potter, and Y. Zhang, Es-timation of probability distributions for individual parameters usingaggregate population data, CRSC-TR98-6, NCSU, January, 1998; InStochastic Analysis, Control, Optimization and Applications, (Editedby W. McEneaney, G. Yin and Q. Zhang), Birkhauser, Boston, 1989.

[BG1] H.T. Banks and N.L. Gibson. Well-posedness in Maxwell systemswith distributions of polarization relaxation parameters. CRSC, TechReport TR04-01, North Carolina State University, January, 2004; Ap-plied Math. Letters, 18 (2005), 423–430.

[BG2] H.T. Banks and N.L. Gibson, Electromagnetic inverse problems in-volving distributions of dielectric mechanisms and parameters, CRSC-TR05-29, August, 2005; Quarterly of Applied Mathematics, 64 (2006),749–795.

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[BGIT] H.T. Banks, S.L. Grove, K. Ito and J.A. Toivanen, Static two-playerevasion-interrogation games with uncertainty, CRSC-TR06-16, June,2006; Comp. and Applied Math, 25 (2006), 289–306.

[BI86] H.T. Banks and D.W. Iles, On compactness of admissible parame-ter sets: convergence and stability in inverse problems for distributedparameter systems, ICASE Report #86-38, NASA Langley Res. Ctr.,Hampton VA 1986; Proc. Conf. on Control Systems Governed byPDE’s, February, 1986, Gainesville, FL, Springer Lecture Notes in Con-trol & Inf. Science, 97 (1987), 130-142.

[BI] H.T. Banks and K. Ito, A unified framework for approximation andinverse problems for distributed parameter systems. Control: Theoryand Adv. Tech. 4 (1988), 73-90.

[BI2] H.T. Banks and K. Ito, Approximation in LQR problems for infinitedimensional systems with unbounded input operators, CRSC-TR94-22,N.C. State University, November, 1994; in J. Math. Systems, Estima-tion and Control, 7 (1997), 119–122.

[BIKT] H. T. Banks, K. Ito, G. M. Kepler and J. A. Toivanen, Materialsurface design to counter electromagnetic interrogation of targets, Tech.Report CRSC-TR04-37, Center for Research in Scientific Computation,N.C. State University, Raleigh, NC, November, 2004; SIAM J. Appl.Math., 66 (2006), 1027–1049.

[BIT] H. T. Banks, K. Ito and J. A. Toivanen, Determination of interrogat-ing frequencies to maximize electromagnetic backscatter from objectswith material coatings, Tech. Report CRSC-TR05-30, Center for Re-search in Scientific Computation, N.C. State University, Raleigh, NC,August, 2005; Communications in Comp. Physics, 1 (2006), 357–377.

[BKap] H.T. Banks and F. Kappel, Transformation semigroups and L1 ap-proximation for size structured population models (with ), LCDS/CCSRep. 88-20, July, 1988; Semigroup Forum, 38 (1989), 141–155.

[BKW1] H.T. Banks, F. Kappel and C. Wang, Weak solutions and differ-entiability for size structured population models (with ), CAMS Tech.Rep. 90-11, August, 1990, University of Southern California; in DPSControl and Applications, Birkhauser, Intl. Ser. Num. Math., Vol.100(1991), 35–50.

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[BKW2] H.T. Banks, F. Kappel and C. Wang, A semigroup formulationof nonlinear size-structured distributed rate population model, CRSC-TR94-4, March 1994; in Control and Estimation of Distributed Param-eter Systems: Nonlinear Phenomena, Birkhauser ISNM, 118 (1994),1–19.

[BKa] H.T. Banks and P. Kareiva, Parameter estimation techniques fortransport equations with application to population dispersal and tis-sue bulk flow models, LCDS Report #82-13, Brown University, July1982; J. Math. Biol., 17 (1983), 253–273.

[BK] H.T. Banks and K. Kunisch. Estimation Techniques for DistributedParameter Systems. Birkhausen, Bosten, 1989.

[BKcontrol] H.T. Banks and K. Kunisch, An approximation theory for non-linear partial differential equations with applications to identificationand control, LCDS Technical Report #81-7, Brown University, April1981; SIAM J. Control & Opt., 20 (1982), 815–849.

[BKurW1] H. T. Banks, A. J. Kurdila and G. Webb, Identification of hys-teretic control influence operators representing smart actuators, Part I:Formulation, Mathematical Problems in Engineering 3 (1997), 287–328.

[BKurW2] H. T. Banks, A. J. Kurdila and G. Webb, Identification of hys-teretic control influence operators representing smart actuators, PartII. Convergent approximations, J. of Intelligent Material Systems andStructures 8 (1997, 536–550.

[BaPed] H.T. Banks and M. Pedersen, Well-posedness of inverse problemsfor systems with time dependent parameters, CRSC-TR08-10, August,2008; Arabian Journal of Science and Engineering: Mathematics, 1(2009), 39–58.

[BPi] H.T. Banks and G.A. Pinter, A probabilistic multiscale approach tohysteresis in shear wave propagation in biotissue, CRSC-TR04-03, Jan-uary, 2004; SIAM J. Multiscale Modeling and Simulation, 3 (2005),395–412.

[BPo] H.T. Banks and L.K. Potter, Probabilistic methods for addressinguncertainty and variability in biological models: Application to a tox-icokinetic model, CRSC-TR02-27, N.C. State University, September,2002; Math. Biosci., 192 (2004), 193–225.

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[BRS] H.T. Banks, K.L. Rehm and K.L. Sutton, Dynamic social networkmodels incorporating stochasticity and delays, CRSC-TR09-11, May,2009; Quarterly Applied Math., to appear.

[BSW] H.T. Banks, R.C. Smith and Y. Wang, Smart Material Struc-tures: Modeling, Estimation and Control, Masson/John Wiley,Paris/Chichester, 1996.

[BSTBRSM] H.T. Banks, Karyn L. Sutton, W. Clayton Thompson, Gen-nady Bocharov, Dirk Roose, Tim Schenkel, and Andreas Meyerhans,Estimation of cell proliferation dynamics using CFSE data, CRSC-TR09-17, August, 2009; Bull. Math. Biology, submitted.

[BT] H.T. Banks and H.T. Tran, Mathematical and Experimental Modelingof Physical and Biological Processes, CRC Press, Boca Raton, FL, 2009.

[Basar] T. Basar and G. J. Olsder, Dynamic Noncooperative Game Theory,SIAM Publ., Philadelphia, PA, 2nd edition, 1999.

[BA] G. Bell and E. Anderson, Cell growth and division I. A mathemati-cal model with applications to cell volume distributions in mammaliansuspension cultures, Biophysical Journal, 7 (1967), 329–351.

[B] P. Billingsley. Convergence of Probability Measures. Wiley, New York,1968.

[BR] G. Birkhoff and G.C. Rota. Ordinary Differential Equations, 1962.

[ButBer] P.L. Butzer and H. Berens, Semi-Groups of Opeerators and Ap-proximation, Springerr-Verlag, New York, 1967.

[CK1] D. Colton and R. Kress. Integral Equation Methods in ScatteringTheory. Wiley, New York, 1983.

[CK2] D. Colton and R. Kress. Inverse Acoustic and Electromagnetic Scat-tering Theory. Springer-Verlag, New York, 1992.

[CH] R. Courant and D. Hilbert. Methods of Mathematical Physics, Vol.1,p. 291.

[D] Dettman. Mathematical Methods in Physics and Engineering, p. 201.

[DS] N. Dunford and J.T. Schwartz. Linear Operators, Vol. I, 1966.

84

[E] R. E. Edwards. Functional Analysis: Theory and Applications. Holt,Rinehart and Winston. New York, 1965.

[Ev] L.C. Evans, Partial Differential Equations, Graduate Studies in Math-ematics, Vol. 19, Amer. Math. Soc., Providence, 2002.

[GP] C. Goffman and G. Pedrick, First Course in Functional Analysis,Prentice-Hall, Englewood Cliffs, NJ, 1965.

[JKH1] J. K. Hale, Functional Differential Equations, Springer-Verlag, NewYork, NY, 1971.

[JKH2] J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, NY, 1977.

[JKH3] J. K. Hale and S. M. Verduyn Lunel, Introduction to FunctionalDifferential Equations, Springer-Verlag, New York, NY, 1993.

[Hal] P. R. Halmos, Measure Theory Springer-Verlag, Berlin-New York, 1974

[HKNT] E. Heikkola, Y. A. Kuznetsov, P. Neittaanmaki and J. Toiva-nen, Fictitious domain methods for the numerical solution of two-dimensional scattering problems, J. Comput. Phys., 145 (1998), 89–109.

[HRT] E. Heikkola, T. Rossi and J. Toivanen, Fast direct solution of theHelmholtz equation with a perfectly matched layer or an absorbingboundary condition, Internat. J. Numer. Methods Engr., 57 (2003),2007–2025.

[HeSt] E. Hewitt and K.R. Stromberg, Real and Abstract Analysis, Springer-Verlag, New York, 1965.

[HP] E. Hille and R. Phillips. Functional Analysis and Semigroups. AMSColloquium Publications, Vol. 31, 1957.

[Hutch] G.E. Hutchinson, Circular causal systems in ecology, Ann. N.Y.Acad. Sci., 50 (1948), 221–246.

[H] P. Huber, Robust Statistics. Wiley, New York, (1981).

[Jackson] J. D. Jackson, Classical Electrodynamics, Wiley & Sons, NewYork, 1975.

[K] T. Kato, Perturbations Theory for Linear Operators, 1966.

85

[Kot] M. Kot, Elements of Mathematical Ecology, Cambridge UniversityPress, Cambridge, 2001.

[KP] M. A. Krasnoselskii and A. V. Pokrovskii, Systems withHysteresis,Springer-Verlag, New York, NY, 1989.

[KS] C. Kravis and J.H. Seinfield, Identification of parameters in distributedparameter systems by regularization, SIAM J. Control and Optimiza-tion, 23 (1985), 217–241.

[Kr] R. Kress. Linear Integral Equations. Springer-Verlag, New York, 1989.

[Li] J.L. Lions, Optimal Control of Systems Governed by Partial DifferentialEquations, Springer-Verlag, New York, 1971.

[LiMag] J.L. Lions and E. Magenes, Non-Homogeneous Boundary ValueProblems and Applications, Vol.I-III, Springer-Verlag, Berlin, Heidel-berg, New York, 1972.

[Ma] I. D. Mayergoyz, Mathematical Models of Hysteresis, Springer-Verlag,New York, NY, 1991.

[McS] E.J. McShane, Integration, Princeton Unoversity Press, Princeton,NJ, 1944.

[MetzD] J. A. J. Metz and E.O. Diekmann, The Dynamics of Physiolog-ically Structured Populations, Lecture Notes in Biomathematics. Vol.68, Springer, Heidelberg, 1986.

[NS] A.W. Naylor and G.R. Sell, Linear Operator Theory in Engineeringand Science, Springer Verlag, New York, 1982.

[Pa] A. Pazy. Semigroups of Linear Operators and Applications to PartialDifferential Equations. Springer-Verlag, New York, 1983.

[P] Yu. V. Prohorov, Convergence of random processes and limit theoremsin probability theory, Theor. Prob. Appl., 1 (1956), 157–214.

[Po] L.K. Potter, Physiologically Based Pharmacokinetic Models for the Sys-temic Transport of Trichloroethylene, Ph.D. Thesis, N.C. State Univer-sity, August, 2001; (http://www.lib.ncsu.edu).

[RM] R.D. Richtmyer and K.W. Morton. Finite Differences Methods forInitial Value Problems, 2d. Edt. Intersciences Publishers, New Tork,1967.

86

[RN] F. Riesz and B.S. Nagy. Functional Analysis. Ungar Press, 1955.

[Ru1] W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, NewYork, 1953.

[Ru2] W. Rudin, Real and Complex Analysis, McGraw-Hill, New York,1966.

[DLR] D.L. Russel, On mathematical models for the elastic beam with fre-quency proportional damping, in Control and Estimation of DistributedParameter Systems, SIAM, Philadelphia, 1992, 125–169.

[SS] J. Sinko and W. Streifer, A new model for age-size structure of a pop-ulation, Ecology, 48 (1967), 910–918.

[Sh] R. E. Showalter. Hilbert Space Methods for Partial Differential Equa-tions. Pitman, London, 1977.

[Smir] V.I. Smirnov, Integration and Functional Analysis, Vol 5 of A Courseof Higher Mathematics,Adison-Wesley, Reading, MA, 1964.

[RCS] R.C. Smith, Smart Material Systems: Model Development, Frontiersin Applied Math, FR32, SIAM, Philadelphia, 2005.

[S] I. Stakgold. Boundary Value Problems of Mathematical Physics, Vol. 1,1967.

[T] H. Tanabe. Equations of Evolution. Pitman, London, 1979.

[Vis] A. Visintin, Differential Models of Hystersis, Springer-Verlag, NewYork, NY, 1994.

[Vog] C.R. Vogel, Computational Methods for Inverse Problems, Frontiersin Applied Math, FR23, SIAM, Philadelphia, 2002.

[VN] J. von Neumann, Zur theorie der gesellschaftsspiele, Math. Ann., 100(1928), 295–320.

[VNM] J. von Neumann and O. Morgenstern, Theory of Games and Eco-nomic Behavior, Princeton University Press, Princeton, NJ, 1944.

[Wahba1] G. Wahba, Bayesian “confidence intervals” for the cross-validatedsmoothing spline, J. Roy. Statist. Soc. B, 45 (1983), 133–150.

87

[Wahba2] G. Wahba, (Smoothing) splines in nonparametric regression, Deptof Statistics-TR 1024, Univ. Wisconsin, September, 2000; In Encyclope-dia of Environmetrics, (A. El-Shaarawi and W. Piegorsch, eds.), Wiley,Vol. 4, pp. 2099–2112, 2001.

[Warga] J. Warga, Optimal Control of Differential and Functional Equa-tions, Academic Press, New York, NY, 1972.

[W] J. Wloka. Partial Differential Equations. Cambridge University Press,Cambridge, 1987.

[Wright] E. M. Wright, A non-linear difference-differential equation, J.Reine Angew. Math., 494 (1955), 66–87.

[YY] Y.S. Yoon and W.W.-G. Yeh, Parameter identification in an inhomo-geneous medium with the finite element method, Soc. Petr. Engr. J.,16 (1976), 217–226.

[Y] K. Yosida. Functional Analysis, 1971.

[1] H. T. Banks, Approximation of nonlinear functional differential equa-tion control systems, J. Optiz. Theory Applic., 29 (1979), 383–408.

[2] H. T.Banks, Identification of nonlinear delay systems using spline meth-ods, Proc. Intl. Conf. on Nonlinear Phenomena in Math. Sciences (V.Lakshmikantham, Ed.), Academic Press, New York, (1982), 47–55.

[3] H. T. Banks, J. E. Banks, L. K. Dick and J. D. Stark, Estimationof dynamic rate parameters in insect populations undergoing sublethalexposure to pesticides, Bulletin of Math Biology, 69 (2007), 2139–2180.

[4] H. T. Banks, J. E. Banks, S. L. Joyner and J. D. Stark, Dynamic modelsfor insect mortality due to exposure to insecticides, Mathematical andComputer Modelling, 48 (2008), 316–332.

[5] H. T. Banks, D. M. Bortz and S. E. Holte, Incorporation of variabil-ity into the mathematical modeling of viral delays in HIV infectiondynamics, Math. Biosciences, 183 (2003), 63–91.

[6] H. T. Banks, D. M. Bortz, G. A. Pinter and L. K. Potter, Model-ing and imaging techniques with potential for application in bioterror-ism, Chapter 6 in Bioterrorism: Mathematical Modeling Applications inHomeland Security, H.T. Banks and C. Castillo-Chavez, eds., Frontiersin Applied Mathematics, SIAM, Philadelphia, 2003, pp. 129–154.

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[7] H. T. Banks, M. Davidian, J. R. Samuels Jr., and K. L. Sutton, Aninverse problem statistical methodology summary, Center for Researchin Scientific Computation Technical Report, CRSC-TR08-01, NCSU,January, 2008; in Mathematical and Statistical Estimation Approachesin Epidemiology, (G. Chowell, et. al., eds.), Springer, New York, toappear.

[8] H. T. Banks and J. L. Davis, A comparison of approximation methodsfor the estimation of probability distributions on parameters, CRSC-TR05-38, NCSU, October, 2005; Applied Numerical Mathematics, 57(2007), 753–777.

[9] H. T. Banks and J. L. Davis, Quantifying uncertainty in the estima-tion of probability distributions with confidence bands, CRSC-TR07-21, NCSU, December, 2007; Math. Biosci. Engr. , 5 (2008), 647–667.

[10] H. T. Banks, S. Dediu and H. K. Nguyen, Sensitivity of dynamicalsystems to parameters in a convex subset of a topological vector space,Math. Biosci. and Engineering, 4 (2007), 403–430.

[11] H. T. Banks and F. Kappel, Spline approximations for functional dif-ferential equations, J. Differential Equations, 34 (1979), 496–522.

[12] J. E. Banks, L. K. Dick, H. T. Banks and J. D. Stark, Time-varyingvital rates in ecotoxicology: selective pesticides and aphid populationdynamics, Ecological Modelling,210 (2008), 155–160.

[13] H. T. Banks and K. Kunisch, Estimation Techniques for DistributedParameter Systems, Birkhauser, Boston, 1989.

[14] H. T. Banks and I. G. Rosen, Spline approximations for linear nonau-tonomous delay systems, J. Mathematical Analysis and Applications,96 (1983), 226–268.

[15] C. C. Carter, T. N. Hunt, D. L. Kline, T. E. Rea-gan and W. P. Barney, Insect and related pests of fieldcrops: pea aphid, Center for Integrated Pest Management,urlℎttp : //ipm.ncsu.edu/AG271/forages/peaapℎid.ℎtml, May,1996.

[16] M. Davidian and D. Giltinan, Nonlinear Models for Repeated Measure-ment Data, Chapman & Hall, London, 1998.

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[17] J. K. Hale, Functional Differential Equations, Springer-Verlag, NewYork, NY, 1971.

[18] J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, NY, 1977.

[19] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Dif-ferential Equations, Springer-Verlag, New York, NY, 1993.

[20] S. L. Joyner, Dynamic Models for Insect Mortality Due to Exposureto Insecticides, M.S. Thesis, North Carolina State University, Raleigh,March, 2008.

[21] M. A. Krasnoselskii and A. V. Pokrovskii, Systems withHysteresis,Springer-Verlag, New York, NY, 1989.

[22] I. D. Mayergoyz, Mathematical Models of Hysteresis, Springer-Verlag,New York, NY, 1991.

[23] G. A. F. Seber and C. J. Wild, Nonlinear Regression, John Wiley &Sons, New York, NY, 1989.

[24] J. D. Stark and U. Wennergren, Can population effects of pesticides bepredicted from demographic toxicological studies? J. Economic Ento-mology, 88, (1995), 1089–1096.

[25] A. Visintin, Differential Models of Hystersis, Springer-Verlag, NewYork, NY, 1994.

[26] J. Warga, Functions of relaxed controls, SIAM J. Control, 5 (1967),628–641.

[27] J. Warga, Optimal Control of Differential and Functional Equations,Academic Press, New York, NY, 1972.

[28] H.T. Banks and J. Burns, Hereditary control problems: numericalmethods based on averaging approximations, SIAM J. Control and Op-timization, 16(2), 1978, pp. 169-208.

[29] H.T. Banks and J. Burns, Introduction to Control of Distributed Pa-rameter Systems, CRSC Lecture Notes, 1996, to appear.

[30] H.T. Banks and D.W. Iles, On compactness of admissible parametersets: convergence and stability in inverse problems for distributed pa-rameter systems, ICASE Report 86-38, 1986; Proc. Conf. on Control

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Systems Governed by PDE’s, Feb., 1986, Gainesville, FL, Springer Lec-ture Notes in Control and Inf. Science, 97, 1987, pp. 120-142.

[31] H.T. Banks and K. Ito, A unified framework for approximation in in-verse problems for distributed parameter systems, Control-Theory andAdvanced Technology, 4, 1988, pp. 73-90.

[32] H.T. Banks and K. Ito, Approximation in LQR problems for infinitedimensional systems with unbounded input operators, CRSC Techni-cal Report CRSC-TR94-22, November 1994; Journal of MathematicalSystems, Estimation and Control, to appear.

[33] H.T. Banks, K. Ito and C. Wang, Exponentially stable approxima-tions of weakly damped wave equations, in Distributed Parameter Sys-tems Control and Applications, (F. Kappel et al, eds.), ISNM Vol. 100,Birkhauser, 1991, pp. 1-33.

[34] H.T. Banks and F. Kappel, Spline approximation for functional differ-ential equations, Journal of Differential Equations, 34, 1979, pp. 496-522.

[35] H.T. Banks and K. Kunisch, The linear regulator problem for parabolicsystems, SIAM J. Control and Optimization, 22, 1984, pp. 684-698.

[36] H.T. Banks, I.G. Rosen and K. Ito, A spline-based technique for com-puting Riccati operators and feedback controls in regulator problemsfor delay systems, SIAM J. Sci. Stat. Comp., 5, 1984, pp. 830-855.

[37] D.S. Bernstein and D.C. Hyland, The optimal projection equationsfor finite-dimensional fixed-order dynamic compensation of infinite-dimensional systems, SIAM J. Control and Optimization, 24(1), 1986,pp. 122-151.

[38] J.A. Burns and H. Marrekchi, Optimal fixed-finite-dimensional com-pensator for Burgers’ equation with unbounded input/output opera-tors, ICASE Report 93-19; Computation and Control III, (K.L. Bowersand J. Lund, eds.), Birkhauser, Boston, 1993, pp. 83-104.

[39] R.F. Curtain, Finite dimensional compensators for parabolic dis-tributed systems with unbounded control and observation, SIAM J.Control and Optimization, 22(2) 1984, pp. 255-276.

91

[40] R.F. Curtain and D. Salamon, Finite dimensional compensators forinfinite dimensional systems with unbounded input operators, SIAM J.Control and Optimization, 24(4), 1986, pp. 797-816.

[41] G. Da Prato, Synthesis of optimal control for an infinite dimensionalperiodic problem, SIAM J. Control and Optimization, 25(3), 1987,pp. 706-714.

[42] J.S. Gibson and A. Adamian, Approximation theory for linear-quadratic-Gaussian optimal control of flexible structures, SIAM J. Con-trol and Optimization, 29(1), 1991, pp. 1-37.

[43] K. Ito, Finite-dimensional compensators for infinite-dimensional sys-tems via Galerkin-type approximations, SIAM J. Control and Opti-mization, 28(6), 1990, pp. 1251-1269.

[44] K. Ito and H.T. Tran, Linear quadratic optimal control problem forlinear systems with unbounded input and output operators: numeri-cal approximations, Proceedings of the 4th International Conference onControl of Distributed Systems, (F. Kappel et al, eds.), ISNM Vol. 91,Birkhauser, 1989, pp. 171-195.

[45] F. Kappel and D. Salamon, Spline approximation for retarded systemsand the Riccati equation, SIAM J. Control and Optimization, 25(4),1987, pp. 1082-1117.

[46] F. Kappel and D. Salamon, On the stability properties of spline ap-proximations for retarded systems, SIAM J. Control and Optimization,27(2), 1989, pp. 407-431.

[47] E. Kreyszig, Introductory Functional Analysis with Applications, WileyClassics Edition, John Wiley and Sons, New York, 1989.

[48] I. Lasiecka, Galerkin approximations of infinite-dimensional compen-sators for flexible structures with unbounded control action, Acta Ap-plicandae Mathematicae, 28, 1992, pp. 101-133.

[49] I. Lasiecka, Finite element approximations of compensator design foranalytic generators with fully unbounded controls/observations, SIAMJ. Control and Optimization, 33(1), 1995, pp. 67-88.

[50] I. Lasiecka, D. Lukes and L. Pandolfi, Input dynamics and nonstan-dard Riccati equations with applications to boundary control of damped

92

wave and plate equations, Journal of Optimization Theory and Appli-cations, 84(3), 1995, pp. 549-574.

[51] I. Lasiecka and R. Triggiani, Numerical approximations of algebraicRiccati equations for abstract systems modelled by analytic semigroups,and applications, Mathematics of Computation, 57(196), 1991, pp. 639-662 and S13-S37.

[52] I. Lasiecka and R. Triggiani, Algebraic Riccati equations arising fromsystems with unbounded input-solution operator: applications toboundary control problems for wave and plate equations, NonlinearAnalysis, Theory, Methods and Applications, 20(6), 1993, pp. 659-695.

[53] D.G. Luenberger, An introduction to observers, IEEE Trans. Auto.Control, AC-16, 1971, pp. 596-602.

[54] A.J. Pritchard and D. Salamon, The linear quadratic control problemfor infinite dimensional systems with unbounded input and output op-erators, SIAM J. Control and Optimization, 25(1), 1987, pp. 121-144.

[55] J.M Schumacher, A direct approach to compensator design for dis-tributed parameter systems, SIAM J. Control and Optimization, 21(6),1983, pp. 823-836.

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