numerical methods for linear impulse feedback problems

207

ISSN 1064-2307, Journal of Computer and Systems Sciences International, 2008, Vol. 47, No. 2, pp. 207–213. © Pleiades Publishing, Ltd., 2008.Original Russian Text © A.N. Daryin, A.Yu. Malakaeva, 2008, published in Izvestiya Akademii Nauk. Teoriya i Sistemy Upravleniya, 2008, No. 2, pp. 50–56.

INTRODUCTION

The problems of minimization of variation of con-trol on the trajectories of a linear controllable system[1–3] arise in control of instantaneous corrections ofmotion in space, in systems with communicational con-straints, and in hybrid systems. The corresponding opti-mal controls have impulse character, i.e., are general-ized functions.

Solutions of problems of impulse control have beenfound basically in the class of open-loop controls [4

−

6].The application of methods of dynamical programming[7, 8] provides an opportunity to present the optimalvalue of variation in the form of solution of quasivaria-tional inequality of Hamilton–Jacobi–Bellman (HJB)type [9] and based on this, to specify the set of states inwhich jump occurs and the values of these jumps, i.e.,to find the impulse control synthesis This approach alsoallows us to solve problems admitting controls in aform of derivatives of generalized functions. (see [2]).

In this paper, numerical algorithms for constructingthe synthesis of impulse controls for a linear system,based on approximation of reachability sets and polarsets are described. The connection of the polar sets withthe value function is shown and the results of numericalsimulation are presented.

1. PROBLEM STATEMENT

Consider the problem of impulse control

(1.1)

(1.2)

Here

x

(

t

)

∈

R

n

is the state vector, which is a function ofbounded variation, continuous from the left and havinga limit from the right,

U

(

·

)

∈

BV

([

t

0

,

t

1

];

R

m

)

is a gener-

J U ·( )( ) U ·( ) inf,t0 t1,[ ]Var=

dx t( ) A t( )x t( )dt B t( )dU t( ),+=

x t0( ) x0, x t1 0+( ) 0.= =

alized control, is the total variation of the

control on the interval

[

t

0

,

t

1

]

, and

BV

([

t

0

,

t

1

];

R

m

)

is thespace of functions of bounded variation with values in

R

m

. The matrix functions

A

(

t

)

∈

R

n

×

n

,

B

(

t

)

∈

R

n

×

m

arecontinuous. We will assume that the function

x

(

t

)

,which is a solution of (1.2), is continuous from the leftfor all

t

. The final time instant

t

1

is fixed. The problemwith the non-zero right end

x

(

t

1

+ 0) =

x

1

, may bereduced to (1.1)–(1.2) by a change of variables.

Problem 1.

Find the control synthesis in problem(1.1)–(1.2), i.e., point out the set

J

⊆

[

t

0

,

t

1

]

×

R

n

ofstates

(

t

,

x

)

in which the control has an impulse and findthe value of this impulse depending on state, i.e., thefunction

v

(

t

,

x

) :

J

R

m

\{0}. Assume that the con-trol synthesis control synthesis does not admit simulta-neous consecutive jumps at the same time instant: if

(

t

,

x

)

∈

J

then for the point

x

' =

x

+

B

v

(

t

,

x

)

,

(

t

,

x

')

∈

/

J

holds.After presenting numerical algorithms for Problem 1, itwill be pointed out in which way they can be applied tothe minimization of a functional of the Mayer-Bolzatype and to the problem with derivatives of generalizedfunctions (see Problems 2 and 3, respectively).

Problem 2.

Find the control synthesis (the

J

andfunction

v

) for the problem of minimization of thefunctional

where the given terminal function

ϕ

(

x

)

is proper,closed, and convex [10].

The detailed problem statement for the system withdistributional derivatives is presented in [2]. We assumethat the matrix functions

A

(

t

),

B

(

t

)

are

k

-fold continu-ously differentiable on the interval

(

α

, β) ⊃ [t0, t1].

Problem 3. Find a control synthesis, i.e., the set ofstates J, in which jumps occur, and the value ofimpulses of each order v0(t, x), v1(t, x), …, vk(t, x) in

U ·( )t0 t1,[ ]Var

J U ·( )( ) U ·( )t0 t1,[ ]Var ϕ x t1 0+( )( ) inf,+=

DISCRETESYSTEMS

Numerical Methods for Linear Impulse Feedback ProblemsA. N. Daryin and A. Yu. Malakaeva

Moscow State University, Vorob’evy gory, Moscow, 119899, RussiaReceived April 18, 2007

Abstract—Numerical algorithms of synthesis of impulse controls for a linear system, based on approximationof reachability sets and polar sets by polyhedra are described. The connection of the polar sets with the valuefunction in the dynamical programming method is shown. The results of numerical simulation are presented.

DOI: 10.1134/S1064230708020068

208

JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL Vol. 47 No. 2 2008

DARYIN, MALAKAEVA

these states for the problem of minimization of thefunctional

under the condition (1.2). Here N* is the conjugatenorm

on the space Dkm[t0, t1] of k-fold differentiable functionswith values in Rm. In this case, if ( ) ∈ J, then in theneighborhood of the instant , the control has the form,

(1.3)

2. DYNAMIC PROGRAMMING METHODRecall some facts we need from [7, 8]. By the value

function V(t0, t1) of Problem 1, we call the optimalvalue of variation of control

It may be presented as [4]

(2.1)

where X(t, τ) is the fundamental matrix, i.e., the solu-tion of the Cauchy problem for the matrix differential

equation = A(t)X(t, τ), X(τ, τ) = I.. The value

function V(t, x) is a viscosity solution of quasivaria-tional inequality of Hamilton–Jacoby–Bellman type,

(2.2)

with the initial condition

Here S1 is a unit sphere in the space Rm centered at 0,the symbol ⊕ means the pseudoinverse matrix. Accordingto (2.2), at any position (t*, x*), we have one of two options:either H1 = 0, and then one may choose dU(t*) = 0, orH1 > 0, then H2 = necessarily, and the control shouldmake a jump in the direction BT(t*)Vx; i.e., the control

J U ·( )( ) N*dUdt-------⎝ ⎠

⎛ ⎞ inf=

N ψ t( )( )

= ψ t( ) 2 ψ ' t( ) 2 … ψ k( ) t( ) 2+ + +( )

12---

t t0 t1,[ ]∈max

t, x

t

dU t( ) v i t x,( )δ i( ) t t–( ).i 0=

k

∑=

V t0 x0,( ) J U ·( )( ) under condition 1.2( ){ }.U ·( )inf=

V t x,( )p X t1 t,( )x,⟨ ⟩

BT ·( )XT t1 ·,( ) p C t0 t1,[ ]

-------------------------------------------------------,p R

m∈

sup=

X t τ,( )∂t∂

-------------------

min H1 t x Vt V x, , ,( ) H2 t x Vt V x, , ,( ),{ } 0,=

H1 t x,( ) H1 t x Vt V x, , ,( ) Vt V x A t( )x,⟨ ⟩ ,+= =

H2 t x,( ) H2 t x Vt V x, , ,( )=

= V x B t( )u,⟨ ⟩u S1∈min 1+ 1 BT t( )V x–=

V t1 x,( ) B⊕ t1( )x , x imB;∈+∞ otherwise.⎩

⎨⎧

=

in a neighborhood of this point has a form dU(t) = –αBT(t*)Vxδ(t – t*). In this case, the value of the jump αis determined from conditions:

3. NUMERICAL ALGORITHMS

Value function (2.1) is positive homogeneous withrespect to the argument x. Hence, it can be presented inthe form of a gage function

V(t, x) = min{α > 0 | α–1x ∈ X1[t]}, (3.1)

where X1[t] is a non-empty convex compact. SinceX1[t] = {x|V(t, x) ≤ 1}, X1[t] is the reachability set ofsystem (1.2) in reverse time from the point x(t1 + 0)under the condition ≤ 1.

here x = x(t; t1 + 0, 0 | U) is the solution of the systemfor control U with the initial condition x(t1 + 0) = 0. Byvirtue of positive homogeneity, the value function maybe also presented in the form of the supporting functionof a nonempty convex compact Z[t]

We will call Z[t] the polar set. The numerical methodsdescribed, are based on approximation of the sets X1[t]and Z[t] by convex polyhedra.

3.1 Reachability Set

The reachability set X1[t] may be presented in theform of convex hull of points, reachable by one impulseof unit amplitude [3]

(3.2)

One may obtain the internal approximation of thereachability set X1[t] by a convex polyhedron by replac-ing in (3.2) the interval [t, t1] by a finite set of points t =τ0 < τ1 < … < τK = t1 and unit sphere {||u|| = 1}, by a finiteset of unit vectors u1, …, uM (in the case of m, we choseM = 2, u1 = –1, u2). Thus, we form the set

Applying to this set the algorithm for constructing theconvex hull [11], we obtain a set of N facets. Each facet

H2 t* x* βBT t*( )V x t* x*,( )–, ,( ) 0, β 0 α,[ ];∈=

H1 t* x* αBT t*( )V x t* x*,( )–, ,( ) 0.=

U ·( )t t1,[ ]

Var

X1 t[ ] X1 t; t1 0,( )=

= x t; t1 0+ 0 U,( ) U s( )( ) 1≤s t t1,[ ]∈

Var{ },

V t x,( ) ρ x Z t[ ]( ) x p,⟨ ⟩ .p Z t[ ]∈sup= =

X1 t[ ] conv X t τ,( )B τ( )u{ }.u 1=∪

τ t t1,[ ]∈∪=

X1 t[ ]= conv X t τ,( )B τ( )u{ } X1 t[ ].⊆

u u0…uM{ }∈∪

τ τ0…τK{ }∈∪


NUMERICAL METHODS FOR LINEAR IMPULSE FEEDBACK PROBLEMS 209

is described by n vertices. linearly independent vectors

…, j = 1, …, N.

Let us describe how to calculate the upper estimatefor the value function

(3.3)

For each facet j of convex hull of the set , wecompose the matrix

An arbitrary vector x ∈ Rn may be decomposed by the

basis …, and the decomposition coefficients arecomputed with the help of the matrix Mj

Find the number j, for which x ∈ cone{ …, } (thecone). This condition is equivalent to the condition λi ≥ 0,

i = 1, …, n. Consider a point y = Λ–1x, Λ = =

⟨1, λ⟩, 1 = (1, …, 1)T.

Since = ⟨1, y⟩ = 1, the point y islocated on the facet j. Hence, V(t, y) = 1. Having usedpositive homogeneity of the value function, we obtain

From this, one may also find the gradient of the func-

tion (at those points, where it exists)

At the position (t, x), the control may have an impulse,if the condition (cf. [8])

holds, and the direction of the impulse is =

−BT(t)( )T1. Find the amplitude of the impulse such that the impulse at the time instant t is equal to u =

For this sake, we will require that, after the jump,the system trajectory would attain the boundary of facet j

The value of can be determined as follows. Let λ =

x, µ = B(t) then

Remark 1. For the purpose of solving the synthesis

problem, it is necessary to compute the set for agiven set of time instants τ0 < τ1 < … < τK. In this case,

x1j , xn

j ,

V t x,( ) min α 0 α 1– x X1 t[ ]∈>{ }.=

X1 t[ ]

M j x1j , …, xn

j( ) Rn n× .∈=

x1j , xn

j ,

x λixij, λ

i 1=

n

∑ M j1– x.= =

x1j , xn

j

λii 1=n∑

M j1– y( )ii 1=

n∑ M j1–

V t x,( ) M j1–( )T

1 x,⟨ ⟩ , M j1– x 0.≥=

V

V x t x,( ) M j1–( )T

1, M j1– x 0.≥=

BT t( )V x 1 ⇔ BT t( ) M j1–( )T

1 1,≥≥

u

M j1– α,

α u.

α max α M j1– x αB t( )u 0≥+( ){ }.=

αM j

1– M j1– u,

α λiµi1– µi 0<–{ }.

i 1, …, n=min=

X1 t[ ]

one may compute the corresponding convex hulls con-secutively, using the representation.

3.2 Polar Set

The polar set Z[t] may be presented in the form (cf.[8])

Set K + 1 moments of time t = τ0 < τ1 < … < τk = t1 andM unit vectors u1, …, uM ∈ Rm (in the case m = 1, wechoose M = 2, u1 = 1, and u2 = –1), then

(3.4)

The set is the exterior estimate of the polar setZ[t]; therefore the function

(3.5)

is the upper estimate for the value function V(t, x). We

use the same notation , that in the previous section,since for the same choice of the points τj, uj estimates

(3.3) and (3.5) coincide. Since set is defined by afinite set of linear inequalities, (3.5) presents a linearprogramming problem.

If the value (t, x) is finite, then

Under the condition, that there exists a unique maxi-mizer p(t, x), the function is differentiable at the

point t, x and (t, x) = p(t, x) [12].The control may also have an impulse in the position

t, x, if

holds, and, in this case, the direction of the impulse = –BT(t)p(t, x). We will find the amplitude of the

impulse as the maximal value α > 0 at which p(t, x) stillremains a maximizer in (3.5) after the jump, i.e.,

The value of can be found in the following way. Letξ1, …, ξn be active linearly independent constraints in

X1 τi 1–[ ] convc--- X τi 1– τi,( )X1 τi[ ]( )⎩⎨⎧

=

∪ B τi 1– u( ){ }u ui, …, uM{ }∈

∪( )⎭⎬⎫

.

Z t[ ] p BT τ( )XT t τ,( ) p 1≤{ }.τ t t1,[ ]∈∩=

Z t[ ] Z t[ ]⊆= p p X t τ,( )B τ( )u,⟨ ⟩ 1≤{ }.

u u0…uM{ }∈∩

τ τ0…τK{ }∈∩

Z t[ ]

V t x,( ) x p,⟨ ⟩p Z t[ ]∈sup=

V

Z t[ ]

V

V t x,( ) x p t x,( ),⟨ ⟩ , p t x,( ) x p,⟨ ⟩ .p Z t[ ]∈

Argmax∈=

V

V xˆ

BT t( )V x 1 ⇔ BT t( ) p t x,( ) 1,≥≥

u

p t x,( ) x αB t( )u+ p,⟨ ⟩ .p Z t[ ]∈

Argmax∈

α

210


DARYIN, MALAKAEVA

problem (3.4), (3.5), i.e., ⟨ξi, p⟩ = 1, i = . Expandvector x by the basis ξi

Since

then its necessary λi ≥ 0, i = 1, …, n. Hence,

Denote µ = Ξ–1B(t) , then

Remark 2. In the course of simulation of the controlprocess, we do not need to solve a linear programmingproblem in (3.5). Suppose that at a time instant τk, thecontrol did not have an impulse and the vector pk = p(τk,x(τk)) is known. Moreover, ⟨pk, B(τk)B(τk)ui⟩ < 1, ∀i =

i.e., all constraints corresponding to the currenttime instant τk, then

(3.6)

If the assumptions made are not satisfied, then the right-hand side of (3.6) is usually a good initial approxima-tion for finding p(τk + 1, x(τk + )).

3.3 Error Bound

Theorem 1. Let matrix functions X(t, τ) and B(t) beLipschitz continuous. Then at m = 1 and as K ∞,

and in the case if m ≥ 2, the points uj can be chosen sothat as K ∞, M ∞ the inequality

will hold. Here h is the Hausdorff metric on the spaceof compact sets, K + 1 is the number of points of thepartitioning of the interval [t, t1], M + 1 is the numberof unit vectors uj.

Proof In fact,

1 n, .

x λiξi, λi 1=

n

∑ Ξ 1– x, Ξ ξ1, …, ξn( ).= = =

x p,⟨ ⟩ λi ξi p,⟨ ⟩ ,i 1=

n

∑=

α max α Ξ 1– x αB t( )u+( ) 0≥{ }.=

u,

α λiµi1– µi 0<–{ }.

i 1, …, n=min=

1 M, ,

p τk 1+ x τk 1+( ),( ) XT τk τk 1+,( ) p τk x τk( ),( ).=

h X1 t[ ] X1 t[ ],( ) O K 1–( ),=

h X1 t[ ] X1 t[ ],( ) O K 1– M2

m 1–-------------–

+⎝ ⎠⎛ ⎞=

h X t τ,( )B τ( )B1 X t τ j,( )B τ j( ) u j{ },( )≤ h X t τ,( )B τ( )B1 X t τs,( )B τs( )B1,( )

+ h X t τs,( )B τs( )B1 X t τs,( )B τs( ) u j{ },( ).

The first term is estimated as O(K–1), since X and B areLipschitz–continuous. If m = 1, then the second term isequal to zero, otherwise it does not exceed

This estimate is proved in [13], and the constructivemethod of optimal choice of the points uj is alsodescribed there.

Remark 3. From the theorem, one should choosethe number of points for approximation consistently,

namely, M ~ Remark 4. Note that, the number of points neces-

sary for achieving the given approximation accuracy Mdepends on the dimension of the control m, rather thanon the dimension of the system n.

Theorem 2. Let conditions of Theorem 1 be satis-

fied, and, in addition, 0 ∈ int . Then for m = 1

and in the case of m ≥ 2

Proof. The statement follows from Theorem 1 andthe following lemma.

Lemma. Let A and B be nonempty convex com-pacts, h(A, B) = ε, and BR(0) ⊆ A ⊆ B, and µA(x), andµB(x), are gauge functions of the sets A and B. Then

(3.7)

Proof. Assume that x is a point on the boundary ofthe set B and y is the closest to x point of the set A. Weassume that x � y (otherwise µA(x) � µB(x) and ine-quality (3.7) holds), π is the plane, passing through theray Ox and the point y. In this plane, we draw the tan-gent l to the ball BR(0), passing through the point y andintersecting the ray Ox at w. Let p be the point of touch-ing, an let q be the closest to x point of the straight line l.The triangles ∆xwq and ∆Owp are similar, whichimplies that

On the other hand, ||w – x|| = ||x|| – ||w||, therefore,

Since the points y and p belong to A, by virtue of con-vexity, the point w also belongs to A, which readilyimplies (3.7).

X t ·,( ) C t0 t1,[ ] B ·( ) C t0 t1,[ ] h B1 u j{ },( )⋅ ⋅

= O M2

m 1–-------------–

⎝ ⎠⎛ ⎞ .

Km 1–

2-------------

.

X1 t[ ].

V t x,( ) V t x,( ) V t x,( ) 1 O K 1–( )+[ ],≤≤

V t x,( ) V t x,( ) V t x,( ) 1 O K 1– M2

m 1–-------------–

+⎝ ⎠⎛ ⎞+ .≤≤

1µA x( )µB x( )-------------- 1

εR---.+≤ ≤

w x–w

-----------------x q–

p---------------- ε

R---.≤=

xw

-------- 1εR---.+≤



In the case if y belongs to the ray Ox, the inequality

holds, since y is the closest point to x. Then (3.7) holdsby virtue of the fact that ||x|| ≤ ||y|| + h(A, B) = ||y|| + ε bythe hypothesis ||y|| ≥ R of the statement.

3.4 The MAYER-BOLZA Problem

In Problem 2, the value function can be presented inthe form [8]

where Z[t] is the polar set, ϕ* is a Fenchel conjugate[10] to the terminal function ϕ. As an upper estimate forthe value function, we will choose

(3.8)

The relationship (3.8) presents a constrained optimiza-tion problem with convex functional and linear con-

straints. The solution of this problem (the value (t, x)and the optimal vector p*) may be found by Newton’smethod.

In the particular case, when the terminal function is

a quadratic form ϕ(x) = ⟨x, Gx⟩ with a positive definite

matrix G, its conjugate function is also a quadratic form

ϕ*(p) = ⟨p, G–1p⟩. Hence, in this case, (3.8) is a qua-

dratic programming problem. Having found the vectorp* = Vx, the control synthesis is constructed accordingto the outline described in Section 2.

3.5. A Problem with Derivativesof Generalized Functions

Let us show the way the algorithm we proposed canbe applied to Problem 3. Taking into account [2, Theo-rem 3.1], we obtain that Problem 3 is equivalent toProblem 1 for the extended system

(3.9)

Here U(·) ∈ BV([t0, t1]; Rm(k + 1)). Applying the algo-rithm of control synthesis for system (3.9), we obtain

µA x( )µB x( )-------------- x

y-------≤

V t x,( ) x p,⟨ ⟩ ϕ* p( )–[ ]p Z t[ ]∈sup ,=

V t x,( ) x p,⟨ ⟩ ϕ* p( )–[ ]p Z t[ ]∈sup .=

V

12---

12---

dx t( ) A t( )x t( )dt B t( )dU t( ),+=

B t( ) L0 t( ) L1 t( ) … Lk t( ) , L0 t( ) B t( ),==

L j t( ) A t( )L j 1– t( ) L j 1–' t( ), j– 1 k, .= =

the set J of states in which a jump occurs, and the func-tion of the jump magnitude v(t, x) : J Rm(k + 1). Wepresent its values in the form

Then vj(t, x) are exactly the desired jump values forProblem 3, and the control is represented in form (1.3).

4. EXAMPLE 1

Consider the problem of steering the system

to the origin with the least control variation. Figure 1shows the set of solvability at the initial time instantt0 = 0. Figure 2 presents for the same time instant thevalues of value function, the Hamiltonians (H1 and H2),and the control synthesis. The values are computed onthe arc of unit circle in the space of variables x1, x2 from0 to π (the positive homogeneity of the value functionallows us to recover the values of designated functionsin all remaining points of the phase space).

v t x,( )v 0 t x,( )

�

v k t x,( )⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

, v j t x,( ) Rm.∈=

dx1 t( ) x2 t( )dt,=

dx2 t( ) tx1 t( )–12---x2 t( )–⎝ ⎠

⎛ ⎞ dt 3tdU t( ),cos+=⎩⎪⎨⎪⎧

t 0 π,[ ]∈

1.5

1.0

0.5

0

–0.5

–1.0

–1.5

x2

x1

–1.5 –1.0 –0.5 0 0.5 1.0 1.5

Fig. 1. Solvability set in example 1.

212


DARYIN, MALAKAEVA

5. EXAMPLE 2.

Consider the problem of damping of oscillations ofa harmonic oscillator by a control force applied via theintegrator

dx1 t( ) x2 t( )dt,=

dx2 t( ) x1 t( )dt– x3 t( )dt,+=

dx3 t( ) dU t( ),=

x 0( ) x0, x 2π 0+( ) 0,= =⎩⎪⎪⎨⎪⎪⎧

t 0 2π,[ ], N*dUdt-------⎝ ⎠

⎛ ⎞ inf.∈

Here we admit controls containing the first deriva-tives of delta-functions. This problem is equivalent toProblem 1 for the following system

The value function was computed by means of thealgorithm described in Subsection 3.1. In Fig. 3, thelevel lines of the price function V(π, x) on the unitsphere until the time instant π are shown. The x3-axis isdirected orthogonally to the plane of the picture. Takinginto account the positive homogeneity and symmetry ofthe value function, the data presented in the picturedetermine the value of value function in the wholespace completely.

The Hamiltonian H2 vanishes everywhere by virtueof quasivariational HJB inequality (2.2), hence the opti-mal control may have a jump in any state (and, hence,it is not unique).

In Fig.4, the relationship between components v0(t, x)and v1(π, x) (as well as on the surface of a unit sphere)can be seen. The delta-function type impulses (v0 = 0)correspond to 1, the δ(1)(t– τ) -type impulses (v1 = 0)correspond to 0, and the combination of impulses of bothtypes correspond to the intermediate values. Figure 5shows the domain of vectors x at which the control hasa jump at the time instant t = π.

dx t( ) Ax t( )dt BdU t( ),+=

A0 1 0

1– 0 1

0 0 0⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

, B0 0

0 1

1 0⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

.= =

1.5

1.0

0.5

0

–0.5

–1.0

–1.50 0.5 1.0 1.5 2.0 2.5 3.0

φ

V(t, x)H1(t, x)H2(t, x)u(t, x)

Fig. 2. Value function, Hamiltonians and the control syn-thesis in example 1.

1.0

0.8

0.6

0.4

0.2

0

–0.2

–0.4

–0.6

–0.8

–1.0–1.0 0 1.00.5–0.5

1.050.994

0.754

0.696

0.6350.5750.515

1.35

1.29

1.231.17

1.110.934

0.874

0.814

x2

x1

1.0

0.8

0.6

0.4

0.2

0

–0.2

–0.4

–0.6

–0.8

–1.0–1.0 0 1.00.5–0.5

x2

x1

0.276

0.457

0.548

0.638

0.729

0.819

0.91

Fig. 3. The values of value function V(π, x) on the unitsphere in example 2.

Fig. 4. The relationship between the control componentsv0(π, x) and v1(π, x) on the unit sphere in example 2.



ACKNOWLEDGMENTSThis work was conducted under the support of the

Russian Foundation for Basic Research, project 06-01-00332; the Program of State Support for Leading Sci-entific Schools of Russian Federation, projectNSh-5344.2006.f; and the Scientific Program “HighSchool Scientific Potential Development”, project2.1.1.1714.

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9. A. Bensoussan and J.-L. Lions, Controle ImpulsionnelEt Inequations Quasi Variationnelles (Dunod, Paris,1982).

10. R. T. Rockafeller, Convex Analysis (Princeton Univer-sity Press, Princeton, 1970; Mir, Moscow, 1973).

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1.0

0.8

0.6

0.4

0.2

0

–0.2

–0.4

–0.6

–0.8

–1.0–1.0 0 1.00.5–0.5

x2

x1

Fig. 5. Vectors of the unit sphere at which control must havea jump at t = π in example 2.

numerical methods for linear impulse feedback problems

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