Set-Valued Analysis 11: 1–8, 2003.© 2003 Kluwer Academic Publishers. Printed in the Netherlands.

1

Towards Fully Discretized Differential Inclusions

G. GRAMMELTechnical University Munich, Centre for Mathematics, M6, Arcisstrasse 21,80290 Munich, Germany. e-mail: grammel@appl-math.tu-muenchen.de

(Received: 22 May 2000; in final form: 15 February 2001)

Abstract. Euler schemes for the calculation of the solution set for a differential inclusion are investi-gated. Under Lipschitz and convexity conditions on the set-valued map the usual O(h)-approximation,h denoting the time step, is preserved, if one uses only boundary points in each step. An O(

√h)-

approximation is achieved, if one uses only extremal points. So, in case that the extremal sets arefinite, a full discretization of the differential inclusion is performed.

Mathematics Subject Classifications (2000): 34A45, 34A60.

Key words: differential inclusion, Euler scheme, extremal point.

1. Introduction

We consider differential inclusions

x(t) ∈ F(x(t)), x(0) = x0, t ∈ [0, 1] (1)

in the Euclidean space Rn. It is well known (see, e.g., [1, 4–6, 11, 12]) that its set

of solutions is approximated by the solutions of the difference inclusion

xk+1 − xk ∈ hF(xk), x(0) = x0, k ∈ {0, 1, . . . , [1/h]}, (2)

provided the set-valued map F is Lipschitz continuous on Rn with compact convex

images in Rn. Furthermore, the approximation is linear in the step size h > 0.

The Lipschitz continuity even can be replaced by one-sided Lipschitz continuitytogether with continuity and boundedness of F , see [4]. In this case the approxi-mation order is debased and strongly depends on the modulus of continuity of F .For discontinuous rigth-hand sides a strengthened version of one-sided Lipschitzcontinuity is introduced in [11].

However, the difference inclusion (2) only provides a semi-discretized Eulerscheme for the calculation of the solution set of (1). The discretization of the right-hand sides F(x) is not touched.

On the other hand, in the context of nonlinear control systems with finite con-trol range, it is known that Euler schemes lead to O(

√h)-approximations of the

trajectory set, compare [10], Corollary 2.4.8, or [8, 9]. So, it is a natural question

2 G. GRAMMEL

to ask, under which conditions Euler schemes based on (possibly finite) subsets ofthe right-hand sides F(x) provide an approximation of the trajectory set.

In the present paper, we investigate the approximation properties of differenceinclusions

xk+1 − xk ∈ hG(xk), x(0) = x0, k ∈ {0, 1, . . . , [1/h]}, (3)

where for x ∈ Rn the set F(x) is the closed convex hull of G(x). In particular

we are interested in G(x) = extF(x), the set of extremal points of F(x), and inG(x) = ∂F (x), the set of boundary points of F(x). Both choices are quite naturaland in some sense minimal. Surely, the set of extremal points is the smallest oneguaranteeing any approximation, whereas the set of boundary points is the smallestone (at least for sets F(x) with nonvoid interior) for which we are able to prove alinear approximation.

2. Main Results

Notation. Let X be a normed space. We denote by C(X) the family of compactsubsets ofX and by CC(X) the family of compact convex subsets ofX. ForA,B ∈C(X) and λ ∈ R we set A + B := {a + b : a ∈ A, b ∈ B} and λA := {λa :a ∈ A}. For a point y ∈ X and a compact subset A ∈ C(X) we define the distanceof y to A by

dist(y,A) := inf{‖y − a‖ : a ∈ A}.For a compact subset A ∈ C(X) we define the ε-neighborhood by

Aε := {y ∈ X : dist(y,A) � ε}.For points x ∈ X we write as usual

Bε(x) := {x}ε .For two compact subsets A,B ∈ C(X) we define the distance of A to B by

dist(A,B) := inf{ε > 0 : A ⊂ Bε}and the Hausdorff distance of A and B by

dH(A,B) := sup{dist(A,B), dist(B,A)}.The notion of Hausdorff distance can be extended to precompact subsets

A,B ⊂ X.

Solution Sets. In order to compare the trajectories of the discrete system (3)with the continuous time solutions of the differential inclusion (1), we define for(xk)k∈{0,1,...,[1/h]} the interpolating curve xh ∈ C([0, 1],Rn) by

xh(t) := xk + t − khh

(xk+1 − xk)

TOWARDS FULLY DISCRETIZED DIFFERENTIAL INCLUSIONS 3

for t ∈ [kh, (k + 1)h]. We denote the closure of the set of all interpolating curvesof the discrete system (3) by SGh (x

0). Note that SGh (x0) ∈ C(C([0, 1]; R

n)) by theArzela–Ascoli Theorem, provided the sets G(x) are uniformly bounded.

By SF (x0) we denote the set of all solutions of the differential inclusion (1),i.e. the set of all absolutely continuous functions x: [0, 1] → R

n, which satisfythe initial condition x(0) = x0 and for almost all times t ∈ [0, 1] the inclusionx(t) ∈ F(x(t)). By Theorem 3.5.2 in [2] we also have SF (x0) ∈ C(C([0, 1]; R

n)),if the set-valued map F : R

n → CC(Rn) satisfies the Lipschitz condition

dH(F (x1), F (x2)) � L‖x1 − x2‖for an L � 0.

THEOREM 1. Let P � 0 and F : Rn → CC(Rn), x �→ F(x) be a set-valued map

with F(x) ⊂ BP (0) for all x ∈ Rn.

(i) Suppose that G(x) = F(x) for x ∈ Rn and that the set-valued map x �→

G(x) is LC with Lipschitz constant L � 0. Then, for h ∈ (0, 1], we can estimate

dist(SGh (x

0), SF (x0))

� PLeLh.

(ii) Suppose that extF(x) ⊂ G(x) ⊂ F(x) for x ∈ Rn and that the set-valued

map x �→ G(x) is LC with Lipschitz constant L � 0. Then, for h ∈ (0, 1], we canestimate

dist(SF (x0), SGh (x

0))

� P eL(3 + 2L+ L2 + n)√h.If only extremal points of the right-hand sides are used for the discretization,

we obtain the following result.

COROLLARY 2. Let P � 0 and F : Rn → CC(Rn), x �→ F(x) be a set-valued

map with F(x) ⊂ BP (0) for all x ∈ Rn. Suppose that the set-valued map x �→

G(x) := extF(x) is LC with Lipschitz constant L � 0.Then, for h ∈ (0, 1], we can estimate

dH(SGh (x

0), SF (x0))

� P eL(3 + 2L+ L2 + n)√h.This Corollary 2 is particularly useful in case that, for each x ∈ R

n, the extremalset extF(x) consists of a finite number of points. Then, a full discretization of thedifferential inclusion (1) is achieved.

If only boundary points of the right-hand sides are used for the discretization,we still have a linear convergence.

COROLLARY 3. Let P � 0 and F : Rn → CC(Rn), x �→ F(x) be a set-valued

map with F(x) ⊂ BP (0) for all x ∈ Rn. Suppose that the set-valued map x �→

G(x) := ∂F (x) is LC with Lipschitz constant L � 0.Then, for h ∈ (0, 1], we can estimate

dH(SF (x0), SGh (x

0))

� P eL(4 + 4L+ 2L2)h.

4 G. GRAMMEL

3. Proofs

LEMMA 4. Let F ∈ CC(Rn) with F ⊂ BP (0) for a P � 0. Suppose that extF ⊂G ⊂ F .

Then we can estimate

dist

(F,

1

k

k∑i=1

G

)� P(n+ 1)

k.

Proof. It is sufficient to consider the case extF = G. Let f ∈ F . Accordingto Caratheodory’s theorem (compare, e.g., [3] for a generalized version) there aree0, . . . , en ∈ extF and λ0, . . . , λn ∈ [0, 1] with

f =n∑i=0

λiei, 1 =n∑i=0

λi.

For k � n+ 1 nothing is to show, since dist(F, extF) � P . So we assume thatk > n+ 1. For i = 0, . . . , n we set

µi := [λik] ∈ N0.

Then k − (n+ 1) <∑ni=0 µi � k. We set m := k −∑n

i=0 µi and define

µi :={µi + 1 for i = 0, . . . , m− 1,µi for i = m, . . . , n.

Then we haven∑i=0

µi = k, µi � 0.

We can estimate∥∥∥∥λiei − µi

kei

∥∥∥∥ � P∥∥∥∥λi − µi

k

∥∥∥∥ � P

k

and hence,∥∥∥∥∥1

k

n∑i=0

µiei − f∥∥∥∥∥ �

n∑i=0

∥∥∥∥µik ei − λiei∥∥∥∥ � (n+ 1)P

k

and the proof is finished. ✷Proof of Theorem 1. (i) Let (xk)k∈{0,...,[1/h]} ⊂ R

n be a discrete trajectory of (3).Then we have x0 = x0 and for lh := [1/h] and l ∈ {0, . . . , lh}

xl+1 = xl + hf (xl),

TOWARDS FULLY DISCRETIZED DIFFERENTIAL INCLUSIONS 5

where f (xl) ∈ F(xl). Now we interpolate and define for t ∈ [lh, (l + 1)h]xh(t) := xl + (t − lh)f (xl).

Obviously the curve xh: [0, 1] → Rn, t �→ xh(t) is Lipschitz continuous with

Lipschitz constant P � 0. Furthermore, for almost all t ∈ [0, 1] it holds

dist(xh(t), F (xh(t))

)� LPh.

We conclude by the Filippov Theorem (compare, e.g., [7] or Theorem 5.3.1in [2]) that there is an x ∈ SF (x0) with

‖x − xh‖∞ � LPheL. (4)

(ii) For h ∈ (0, 1] we choose nh ∈ N (to be specified later) and define lh :=[1/(hnh)].

Let x ∈ SF (x0) be a trajectory of the differential inclusion (1). Then we havefor l ∈ {0, . . . , lh}

dist

(1

hnh

∫ (l+1)hnh

lhnh

x(t) dt, F (x(lhnh))

)� LPhnh.

We choose vl ∈ F(x(lhnh)) such that∥∥∥∥ 1

hnh

∫ (l+1)hnh

lhnh

x(t) dt − vl∥∥∥∥ � LPhnh.

We define a family (ηl)l∈{0,...,lh} ⊂ Rn by η0 := x0 and

ηl+1 := ηl + hnhvl.Then we have for l ∈ {0, . . . , lh}

‖ηl − x(lhnh)‖ � LPhnh. (5)

We define a family (ξl)l∈{0,...,lh} ⊂ Rn by ξ0 := x0 and

ξl+1 := ξl + h(l+1)nh−1∑k=lnh

gk(ξl),

where according to Lemma 4, for k = lnh, . . . , (l + 1)nh − 1, the gk(ξl) ∈ G(ξl)are chosen in such a way that∥∥∥∥∥ 1

nh

(l+1)nh−1∑k=lnh

gk(ξl)− vl∥∥∥∥∥ � L‖ξl − x(lhnh)‖ + P(n+ 1)

nh

� L‖ξl − ηl‖ + L‖ηl − x(lhnh)‖ + P(n+ 1)

nh

� L‖ξl − ηl‖ + L2Phnh + P(n+ 1)

nh.

6 G. GRAMMEL

We conclude

‖ξl+1 − ηl+1‖ � ‖ξl − ηl‖ + hnh(L‖ξl − ηl‖ + L2Phnh + P(n+ 1)

nh

)

= ‖ξl − ηl‖(1 + hnhL)+ hnh(L2Phnh + P(n+ 1)

nh

).

By induction we obtain for l ∈ {0, . . . , lh}

‖ξl − ηl‖ � hnh

(L2Phnh + P(n+ 1)

nh

) l−1∑i=0

(1 + hnhL)i

� hnh

(L2Phnh + P(n+ 1)

nh

)1

hnh(1 + hnhL)1/(hnh)

�(L2Phnh + P(n+ 1)

nh

)eL. (6)

With the constructed sequence (ξl) we define a trajectory (xk)k∈{0,...,[1/h]} of thediscrete system (3) by

xk+1 = xk + hg(xk).Here we choose, for k = lnh, . . . , (l + 1)nh − 1, the g(xk) ∈ G(xk) such that

‖g(xk)− gk(ξl)‖ � L‖xk − ξl‖.Then, considering

‖xk − ξl‖ � ‖xk − xlnh‖ + ‖xlnh − ξl‖,we can estimate

‖x(l+1)nh − ξl+1‖ � ‖xlnh − ξl‖ + h(l+1)nh−1∑k=lnh

L(‖xlnh − ξl‖ + hnhP ).

We conclude

‖x(l+1)nh − ξl+1‖ � ‖xlnh − ξl‖(1 + hnhL)+ (hnh)2PL.Since lh � 1/(hnh), we can prove by induction that

‖xlnh − ξl‖ � hnhPLeL (7)

for all l ∈ {0, . . . , lh}. Considering (5), (6) and (7) we obtain

‖x(lhnh)− xlnh‖ � LPhnh + L2PhnheL + P(n+ 1)

nheL + hnhPLeL

TOWARDS FULLY DISCRETIZED DIFFERENTIAL INCLUSIONS 7

and, hence,

‖xh(t)− x(t)‖ � LPhnh + L2PhnheL + P(n+ 1)

nheL +

+ hnhPLeL + 2Phnh

for all t ∈ [0, 1]. Setting nh := [1/√h] we obtain

‖xh − x‖∞ � P(L+ L2eL + (n+ 1)eL + LeL + 2)√h.

We conclude that

dist(SF (x0), SGh (x

0))

� P eL(3 + 2L+ L2 + n)√hand the proof is finished. ✷

Proof of Corollary 2. We just have to observe that the LC of x �→ extF(x)implies the LC of x �→ F(x) with the same Lipschitz constant. The statement thenimmediately follows from Theorem 1. ✷LEMMA 5. For F ∈ CC(Rn), n � 2, we can write

F = 1

2

2∑i=1

∂F = 1

2(∂F + ∂F ).

Proof. Let f ∈ F . If f ∈ ∂F , then nothing is to show. Let f ∈ intF . Considerthe unit sphere Sn−1 ⊂ R

n. For any s ∈ Sn−1 there is a unique γ (s) ∈ ∂F withγ (s)− f = ‖γ (s)− f ‖s. Furthermore, the map s �→ γ (s) is continuous. Hence,there is an s ∈ Sn−1 such that ‖γ (s) − f ‖ − ‖γ (−s) − f ‖ = 0 and thus f =12(γ (s)+ γ (−s)). ✷

Proof of Corollary 3. Firstly, we remark that the LC of x �→ ∂F (x) impliesthe LC of x �→ F(x) with the same Lipschitz constant (and vice versa), see [13].Secondly, for n � 2, we have by Lemma 5 that

dist

(F(x),

1

2

2∑i=1

∂F (x)

)= 0.

Hence, we just have to rewrite part (ii) of the proof of Theorem 1, replacing theestimate (6) by

‖ξl − ηl‖ � (L2Phnh + 0)eL

and setting nh = 2. The case n = 1 is trivial. ✷

8 G. GRAMMEL

References

1. Artstein, Z.: First order approximations for differential inclusions, Set-Valued Anal. 2 (1994),7–18.

2. Aubin, J. P.: Viability Theory, Birkhäuser, Boston, 1991.3. Barany, I.: A Generalization of Caratheodory’s theorem, Discrete Math. 40 (1982), 141–152.4. Donchev, T. and Farkhi, E.: Stability and Euler approximation of one-sided Lipschitz differen-

tial inclusions, SIAM J. Control Optim. 36 (1998), 780–796.5. Dontchev, A. L. and Farkhi, E. M.: Error estimates for discretized differential inclusions,

Computing 41 (1989), 349–358.6. Dontchev, A. L. and Lempio, F.: Difference methods for differential inclusions: A survey, SIAM

Rev. 34 (1992), 263–294.7. Filippov, A. F.: Classical solutions of differential equations with multi-valued right-hand sides,

SIAM J. Control Optim. 5 (1967), 609–621.8. Grammel, G.: Limits of nonlinear discrete time control systems with fast subsystems, Systems

Control Lett. 36 (1999), 277–283.9. Haeckl, G.: Numerical approximation of reachable sets and control sets, Random Comput.

Dynamics 1 (1993), 371–394.10. Haeckl, G.: Reachable sets, control sets and their computation, Thesis, Augsburger

Mathematisch-Naturwissenschaftliche Schriften, Wissner-Verlag, 1996.11. Lempio, F.: Euler’s method revisited, Proc. Steklov Inst. Math. 211 (1995), 429–449.12. Lempio, F. and Veliov, V.: Discrete approximations of differential inclusions, Bayreuth. Math.

Schr. 54 (1998), 149–232.13. Monteiro Marques, M. D. P.: Sur la frontiere d’un convexe mobile, Atti Accad. Naz. Lincei

Rend. Cl. Sci. Fis. Mat. Natur. Ser. VIII 77 (1984), 71–75.