Approximation of the Viability Kernel

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1 Approximation of te Viability Kernel Patrick Saint-Pierre CEREMADE, Université Paris-Daupine Place du Marécal de Lattre de Tassigny Paris cedex october 1990 Abstract We study recursive inclusions x n+1 G(x n ). For instance suc systems appear for discrete finite difference inclusions x n+1 G (x n ) were G := 1 + F. Te discrete viability kernel of G, i.e. te largest discrete viability domain, can be an internal approximation of te viability kernel of K under F. We study discrete and finite dynamical systems. In te Lipscitz case we get a generalization to differential inclusions of Euler and Runge-Kutta metods. We prove first tat te viability kernel of K under F can be approaced by a sequence of discrete viability kernels :associated wit Γ (x) = x + F (x) + Ml 2 2 B. Secondly, we sow tat it can be approaced by finite viability kernels associated wit Γ α (x) := x + F (x) : xn+1 (Γ (x n ) + α()b) X. 1 Introduction Let X a finite dimentional vector space and K a compact subset of X. Let us consider te differential inclusion: { ẋ(t) F (x(t)), for almost all t 0, (1) x(0) = x 0 K, were F is a Marcaud map 1 defined from K to X. Wit tis inclusion, for a fixed > 0, we associate te discrete explicit sceme: { x n+1 x n F (x n ), for all n 1, (2) x 0 = x 0 K, We note G te set-valued map G = 1 + F and te system (2) can be rewrited as follows: (3) x n+1 G (x n ), for all n 0, 1 A set-valued map F : X Y is a Marcaud map if Dom(F ) F is upper-semicontinuous, convex compact valued x Dom(F ), F (x) := max y F (x) y c( x + 1) 1

2 Te Viability Teory allows to study viable solutions of (1) and te subset of elements x 0 K suc tat tere exists at least a viable solution starting at x 0. On te oter and, we look for approximation of suc solutions and we wonder ow te set of initial points from wic tere exists at least a viable approximation solution to (2) and te set of initial points from wic tere exists at least a viable solution to (1) are related togeter. Tese sets are called viability kernel of K under F or discrete viability kernel of K under G. Byrnes & Isidori [5] and Frankowska & Quincampoix [8] ave proposed algoritms wic approximate te viability kernel of K under F wen F is lipscitzian and K is closed. We prove tat, wen F is a Marcaud map, for a good coice of discretizations G, te sequence of discrete viability kernels of K under G converges to a subset contained in te viability kernel of K under F. Moreover it converges to te viability kernel if F is lipscitzian. We sow tat similar results remain true wen we introduce a discretization of te space and consider finite viability kernels. 2 Definitions and General Results We call discrete dynamical system associated wit G te following system: (4) x n+1 G(x n ), for all n 0, We denote by - K te set of all sequences from IN to K. - x := (x 0,..., x n,...) X a solution to discrete dynamical system (4) - S G (x 0 ) te set of solutions x X to te discrete dynamical system starting at x 0. A solution x is viable if and only if x S G (x) K: (5) x n+1 G(x n ), n 0, x 0 = x K x n K, n 0. It means tat tere exists a selection of equation (2) wic remains in K at eac step n. We study te subset of initial points in K from wic tere exists at least one viable solution. Definition 2.1 Let G : X X be a set-valued map. A subset D X is a discrete viability domain of G if (6) x D, G(x) D Let K be a subset of X. Te discrete viability kernel of K under G is te largest closed discrete viability domain contained in K and we denote it V iab G (K). 2

3 We can point out te following remark and properties: V iab G (K) = {x K, suc tat S G (x) K } Since V iab G (K) is te largest discrete viability domain contained in K, any solution of (5) starting from any initial point x 0 K\V iab G (K) never meets te discrete viability kernel V iab G (K) wile it remains in K. Moreover any solution of (5) wic does not start from V iab G (K) must leave K in a finite number of steps. For all closed K 1, K 2 suc tat K 1 K 2 X, ten (7) V iab G (K 1 ) V iab G (K 2 ) X For all setvalued maps G 1, G 2 suc tat x K: G 1 (x) G 2 (x), ten (8) V iab G1 (K) V iab G2 (K) X For all subset K suc tat V iab G (K) K K, ten (9) V iab G (K ) = V iab G (K) 2.1 A Construction Metod for Discrete Viability Kernel Let us consider te sequence of subsets K 0 = K, K 1,..., K n,... defined as follows: K n+1 := {x K n suc tat: G(x) K n } We note K := + K n Proposition 2.1 Let G: X X a upper semicontinuous set-valued map wit closed values and K a compact subset of Dom(G). Ten n=0 (10) K = V iab G (K) Proof Let us prove tat n IN, V iab G (K) K n. We ave V iab G (K) K 0. Since G is upper semicontinuous, K 1 = {x K 0, G(x) K 0 } is closed and, for all x K 0 \K 1, it does not exit any viable solution starting from x. Tis implies recursively tat V iab G (K) (K 0 \K 1 ) = and ten: V iab G (K) K 1 K. 3

4 Let us assume tat Since V iab G (K) = V iab G (K n 1 ) K n 1. K n = {x K n 1, G(x) K n 1 }, for all x K n 1 \K n, it does not exist any solution starting from x viable in K n 1, and tus in K. Ten V iab G (K) (K n 1 \K n ) = and V iab G (K) = V iab G (K n ) K n. Tis implies tat V iab G (K) K. Conversely, from definition 2.1, K is a viability domain: indeed for any x K, n IN : x K n+1 and ten G(x) K n. Since for any fixed x, sets G(x) K n form a decreasing sequence of non empty compact subsets, ten G(x) K is non empty. We ave proved tat K is a viability domain of G and since V iab G (K) is te largest viability domain: K V iab G (K). Definition 2.2 Let G : X X a set-valued map and r > 0. We call extension of G wit a ball of radius r te set-valued map G r : X X defined by : (11) G r (x) := G(x) + rb We consider te sequence of subsets K r,0 = K, K r,1,..., K r,n,... defined as follows: (12) K r,n+1 := {x K r,n suc tat G r (x) K r,n }, K r, := + K r,n If G is an upper semicontinuous set-valued map, G r : X X is also upper semicontinuous and from Proposition 2.1: n=0 (13) r > 0, K r, = V iab G r(k) Wen r decreases to 0, te viability kernel of K under G r converges to te viability kernel of K under G: Proposition 2.2 Let G be upper semicontinuous and K a compact subset of X. Te following property olds: (14) V iab G (K) = r>0 V iab G r(k) 4

5 Proof Let x 0 r>0 V iab Gr(K). For all r > 0, Proposition 2.1 implies: G r (x 0 ) V iab G r(k), r > 0 G r (x 0 ) is closed, V iab G r(k) is compact and bot are, from (8), decreasing sets wen r decreases to zero. Also te intersection G r (x 0 ) V iab G r(k) is a decreasing sequence of nonempty compact sets and r>0(g r (x 0 ) V iab G r(k)) = G(x 0 ) ( V iab G r(k)) r>0 Ten r>0 V iab G r(k) is a viability domain of G. Since V iab G(K) is te viability kernel of G, from definition 2.1 it contains r>0 V iab G r(k). Wen G is a k-lipscitz setvalued map, we ave te following result giving an estimation of te growt of te discrete viability kernel wen r increases: Proposition 2.3 Let G : X X a k-lipscitz set-valued map, K a closed subset of X. Let G r := G + rb, V iab G r(k) and V iab G (K) te discrete viability kernel of G r and G respectively. Ten: (15) V iab G (K) + r k B V iab Gr(K), r > 0 (see footnote 2 ) Proof Let r > 0 given, x V iab G (K) arbitrairely coosen, η < min(η 0, r k ) and x ({x} + ηb) K. Ten (16) { i) x V iabg (K) G(x) V iab G (K) ii) x V iab G r(k) G r (x) V iab G r(k) Since G is k-lipscitz and kη < r, From (16), we deduce tat and since V iab G (K) V iab G r(k) G(x) G(x ) + k x x B G r (x ) G r (x ) V iab G (K) G r (x ) V iab G r(k) Ten x V iab G (K), x ({x} + ηb) K, tere exists a viable solution for te system associated wit G r starting from x and tus x V iab G r(k). 2 wit te convention: + ηb = 5

6 3 Approximation of Viability Kernels for Finite Difference Inclusions Let F a Marcaud map and Γ a sequence of setvalued maps wic correspond to discretizations associated wit te initial differential inclusion (1) satisfying: ( ) Γ 1 (17) ɛ > 0, ɛ > 0, ]0, ɛ ] : Grap Grap(F ) + ɛb were B is te unit ball in X X. We note F := Γ 1. Assumption (17) implies tat te grap of F contains te grapical upper limit 3 of F, tat is to say tat Grap(F ) contains te Painlevé-Kuratowski upper limit 4 of Grap(F ): (18) lim sup Grap(F ) Grap(F ) 0 Let K a sequence of subsets of X suc tat K = lim sup >0 K. Possible K may be constant. Let V iab Γ (K ) te discrete viability kernel of K under Γ. 3.1 Te Viability Kernel Convergence Teorem Teorem 3.1 Let F a Marcaud ) map and Γ a sequence of set-valued maps. Ten te upper limit K = lim sup 0 V iab Γ (K ) suc tat F = CoLim 0 is a viable subset under F : ( Γ 1 lim sup V iab Γ (K ) V iab F (K) 0 Proof - Let us consider x 0 K. Tere exists a subsequence x,0 V iab Γ (K ) wic converges to x 0 and a K -viable solution x := (x 0,..., x n,...) S Γ (x 0 ) K to te discrete system associated wit Γ. From te definition of Γ, x n+1 Γ (x n ) and ten: n > 0, x n+1 x n F (x n ) 3 Te grapical upper limit is te upper limit of te sequence of Grap(F ). 4 Te upper limit of a sequence of subsets D n of X is D = lim sup n D n := {y X lim inf d(y, Dn) = 0} n 6

7 Wit tis sequence we associate te piecewise linear interpolation x ( ) wic coincides to x n at nodes n: Ten We ave x (t) = x n + xn+1 x n (t n), t [n, (n + 1)[, n > 0 ẋ (t) F (x n ), t [n, (n + 1)[ d((x (t), ẋ (t)), Grap(F )) x (t) x n F (x n ) Since F is Marcaud, and from (18), set-valued maps F satisfy a uniform linear growt: c > 0, F (x) c( x + 1), x X As in te proof of te Viability Teorem (see [2],[9]), tis implies { t > 0, x (t) ( x 0 ) + 1)e ct for almost all t > 0, x (t) c( x 0 + 1)e ct Ten, ɛ > 0, t > 0, tere exists ɛ,t > 0 suc tat ]0, ɛ,t ], d((x (t), ẋ (t)), Grap(F )) c( x 0 + 1)e ct ɛ 2 and wit (17) we ave ɛ > 0, ]0, ɛ] : Grap(F ) Grap(F ) + ɛ 2 B Let 0 ɛ,t := min( ɛ, ɛ,t ) > 0 ten (x (t), ẋ (t)) Grap(F ) + ɛb, ]0, ɛ,t ] By te Ascoli and Alaoglu Teorems, we derive tat tere exists x( ) W 1,1 (0, + ; X; e ct dt) and a subsequence (again denoted by) x wic satisfy: { i) x ( ) converges uniformly to x( ) (19) ii) x ( ) converges weakly to x ( ) in L 1 (0, + ; X; e ct dt) Tis implies (see [1] Te Convergence Teorem) tat x( ) is a solution to te differential inclusion: { ẋ(t) F (x(t)), for almost all t 0 x(0) = x 0 K It remains to prove tat te limit is a viable solution: t > 0, tere exists a sequence n t = E( t ) suc tat n t t wen 0. Ten x(t) = lim 0 x (n t ). Since : x (n t ) = x nt V iab Γ (K ), x(t) belongs to te upper limit K of subsets V iab Γ (K ) K and ten K K. Tis implies tat K V iab F (K). 7

8 3.2 Examples of Approximation Processes 1 - Te finite difference explicit sceme. Naturally, te discrete explicit sceme (2) { x n+1 x n F (x n ), for all n 1, x 0 = x 0 K, is associated wit F = F, K = K and G := 1 + F. It already satisfies property (17) for any ɛ. and (20) 2 - Te set-valued Runge-Kutta metod. Let us define te set-valued Runge-Kutta sceme Γ RK : For any x K, F RK (x) := {y X y = 1 2 (β + γ) β were β F (x), γ F (x + β)} 2 Γ RK (x) := 1 + F RK (x) Let (x, y) Grap(Γ RK ). From definition (20), tere exist β F (x) and γ F (x + β)} suc tat y = 1 2 (β + γ) 2 β. Since F (x) is Marcaud, β is bounded by m and since F is upper semicontinuous, ɛ > 0, ɛ,m, ]0, ɛ,m ], F (x + β) F (x) + ɛb. Since F is convex valued, 1 2 (β + γ) F (x) + ɛ 2 B. ɛ RK Ten, coosing min( ɛ, 2m ), we ave F (x) F (x) + ɛb and ten ( ) Γ RK (x) 1 Grap Grap(F (x)) + ɛb Ten condition (17) olds and from Teorem 3.1 we deduce te following corollary: Corollary 3.1 Te upper limit of V iab Γ RK (K) is a viable subset under F : lim sup 0 V iab Γ RK (K) V iab F (K) 3 - Te tickening process. Let us define te set-valued map F T : X X by a tickening of te values of F by balls of radius Ml 2 : (21) F T (x) = F (x) + Ml 2 B 8

9 and consider te set-valued map associated wit te finite differnece sceme for F T : (22) Γ T (x) = x + F T (x) Wen F is Marcaud, we ave te following relations between V iab G (K), V iab Γ T (K) and V iab F (K) : Corollary 3.2 Let F a Marcaud map, G and Γ defined by (22). Ten (23) lim sup 0 V iab G (K) lim sup V iab Γ T (K) V iab F (K) 0 Proof - Te first inclusion olds true since G (x) Γ T (x) and (8). On te oter and, since ( ) Γ T 1 Grap = Grap(F T ) Grap(F ) + Ml 2 B Teorem 3.1 implies te second inclusion. 3.3 Approximation of te Viability Kernel in te Lipscitz case From now on we use te following notations : (24) G = 1 + F F = F + Ml 2 B Γ = 1 + F = 1 + F + Ml 2 2 B Wen F is l-lipscitz, we claim tat te discrete viability kernel V iab Γ (K) is a good approximation of te viability kernel of K under F. Teorem Let F a Marcaud and l-lipscitz set-valued map, let K a closed subset of X satisfying te boundedness condition (25) M := sup sup x K y F (x) y < Ten (26) lim sup V iab Γ (K) = V iab F (K) 0 5 Tis result is due to M. Quincampoix and te autor wen tey visit IIASA Institute - Laxenburg, Austria 9

10 Proof - Since F is Marcaud, from Corollary 3.2, (27) lim sup V iab Γ (K) V iab F (K) 0 We want to ceck te opposite inclusion. Let x 0 K and consider any solution x( ) S F (x 0 ). Let > 0 given. We ave x(t + ) x(t) = t+ ẋ(s) F (x(s)) and F Lipscitzian imply tat x(t + ) x(t) F (x(t)) + l t t+ t ẋ(s)ds, t > 0 x(s) x(t) dsb, t > 0 But since F is bounded, x(s) x(t) (s t)m and tus (28) x(t + ) x(t) F (x(t)) + Ml 2 2 B So, we ave proved tat if x( ) S F (x 0 ) ten te following sequence (29) ξ n = x(n), n 0 is a solution to te discrete dynamical system associated wit Γ : (30) ξ n+1 Γ (ξ n ), n 0 Moreover, if x( ) is a viable solution, ten (ξ n ) n is a viable solution to (30). Tus (31) V iab F (K) V iab Γ (K), > 0 and ten (32) V iab F (K) lim sup V iab Γ (K) Approximation by Finite Setvalued Maps Wit any IR we associate X a countable subset of X, wic spans X in te sense tat (33) x X, x X suc tat x x α() were α() decreases to 0 wen 0: (34) lim α() =

11 4.1 Approximation of discrete and finite viability kernels Let G : X X a finite set-valued map and a subset K Dom(G ). We call finite dynamical system associated wit G te following system: (35) x n+1 G (x n ), for all n 0, and we denote by - K te set of all sequences from IN to K. - x := (x 0,..., xn,...) X a solution to system (35) - S G (x 0 ) te set of solutions x X to te finite differential inclusion (35) starting from x 0 A solution x is viable if and only if x S G (x ) K, tat is to say tat: (36) G (x n ), n 0, x 0 = x K x n K, n 0. x n+1 Let K 0 = K, K 1,..., Kn,... defined recursively as in te second section: K n+1 := {x K n suc tat: G (x ) K n } Te viability kernel algoritm and Proposition 2.1 olds true for finite dynamical systems wenever te set-valued map G as nonempty values and we ave: (37) Let us notice tat K integer p suc tat: V iab G (K ) = K := + K n n=0 can be emptyset and in any case tere exists a finite K = K n = K p, n > p Wat appen wen G is te reduction to K of a set-valued map G? We cannot apply no longer more Proposition 2.1 since G(x ) may not contain any point of te reduction X of X and G (x ) be empty. To turnover tis difficulty, we will consider greater set-valued maps G r wic still approximate G. But te coice of suc approximations is subjet to two opposite considerations: on one and, tey ave to be large enoug in order tat te reductions to X of suc approximations ave teir domain containing K (ave nonempty values on K ), and so, it will be possible to apply again Proposition 2.1. On te oter and, te enlargement is limited as far as te grapical assumption (1) of Teorem 3.1 still olds so as to te viability kernel of K under G contains te upper limit of finite viability kernels of K under te finite set-valued approximations G r. In case of upper semicontinuous set-valued maps, we bring in te fore some discretization process wic leads to approac a subset of a te viability kernel. In te Lipscitz case, tese process enables us to approac te viability kernel completely. 11

12 Notations: te reduction to te finite subset X of any subset D will be noted by a lower index : D := D X ; te extension of a set-valued map G wit a ball of radius r by an upper index: x X, G r (x) := G(x) + rb. Tus te reduction to X of te extension of a set-valued map G wit a ball of radius r will be noted G r. Let us notice tat te extension operation as to be done before te reduction one oterwise it could be empty even for r > α(). From property (33) wic defines α(), we consider now te extension wit r = α(). We observe tat G α() satisfies te non emptyness property: (38) x Dom(G) X, G α() (x ) := G α() (x ) X and te decreasing sequence of finite subsets K α(),0 defined by K α(),n+1 satisfies property (37): := {x K α(),n + K α(), := n=0 = K, K α(),1,..., K α(),n,... suc tat G α() (x) K α(),n } K α(),n = V iab α() G (K ) As a partial conclusion, we are able to approximate te discrete viability kernel of K under G: first we extend G suc tat for all x K, images of G r encounters X, in oter words suc tat Dom(G r ) = Dom(G) X. To be sure of tis, witout loss of generality, we can coose r = α(). Secondly we look after te discrete viability kernel of K under G r, te finite viability kernel of K under G r and at last we let decreasing to 0. Wat relations link togeter te discrete viability kernels V iab G (K) or V iab G α()(k) and te finite viability kernels V iab G (K ) or V iab α() G (K ) wenever K is te reduction of K to X : are te latters te reduction to X of te formers? Does te upper limit of te latters, wen goes to 0, coincide wit te former? 4.2 Properties of te finite viability kernel A first answer is given by applying Proposition 2.2: since lim 0 α() = 0, we ave V iab G α()(k) = V iab G (K) >0 12

13 Te following result gives a necessary and sufficient condition for V iab α() G (K ) to be te reduction of V iab G α()(k) to X : Let G α() : X X, G α() : X X and K a finite subset of Dom(G α() ) defined as follows: G α() (x) := G(x) + α()b K := K X x Dom(G) X : G α() (x ) := G α() (x ) X From definition of α(), x K, G α() (x ). Proposition 4.1 Let G: X X an upper semicontinuous set-valued map wit closed values and K a closed subset of Dom(G). Let r suc tat x Dom(G r ) X, G r (x) X : (39) Dom(G r ) = Dom(G r ) X Ten (40) V iab G r (K ) V iab G r(k) X It coincides if and only if V iab G r(k) X is a discrete viability domain of K under G r : (41) x V iab G r(k) X, G r (x ) (V iab G r(k) X ) Proof From (37) we ave to ceck tat te two following statements (42) x V iab G r(k) X, G r (x ) (V iab G r(k) X ) and (43) V iab G r (K ) = V iab G r(k) X are equivalent. Assume tat (42) olds. Let x V iab G r (K ). Tere exists x S G r (x ) K viable in K K. Since G r (x ) G r (x ), x S G r(x ) K is viable in K. Ten x V iab r G (K) and we obtain inclusion: (44) V iab G r (K ) V iab G r(k) X. On te oter and, V iab G r(k) X is a discrete viability domain of G r. Ten from definition 2.1 and definition of G r it is also a finite viability domain of G r contained in K and tus is contained in te finite viability kernel of K under G r : V iab G r(k) X V iab G r (K ). We obtain te opposite inclusion and prove tat (43) is true. Conversely, if (43) olds, V iab G r(k) X is te finite viability kernel of K under G r, it is obviously a finite viability domain of Gr. Remarks 13

14 1 - Inclusion (40) is always true. Let call A := {x V iab G r(k) X, x V iab G r } and B := {x V iab G r(k) X, G r (x ) (V iab G r(k) X ) = }. 2 - It is easy to prove tat B A and Proposition 4.1 says tat if B is empty, A is empty too. 3 - If A, ten all solutions x to te finite dynamical system starting from any point x 0 A, must leaves K after a finite number of steps, altoug x 0 belongs to te dicrete viability kernel of K under G r. 4 - If B, ten all solutions x to te finite dynamical system starting from any point x 0 B, leaves K at te first step. 5 - If x 0 A and xn is te last element of a solution to te finite dynamical system, starting at x 0, wic is still in K, ten x n B. 6 - If G r (x) K, x K ten for all x K, G r (x ) K. Ten inclusion (40) becomes an equality : K = V iab G r (K ) = V iab G r(k) X. 4.3 Approximation of te viability kernel of K under F by finite viability kernel in te Lipscitz Case Wen G is a k-lipscitz set-valued map, we cannot prove tat in (40), te inclusion becomes an equality. Neverteless we ave in te Lipscitz case an immediate and interesting information about points x K wic do not satisfy (41): Proposition 4.2 Let G : X X a k-lipscitz set-valued map. max(k, 1)α(). For all x V iab G r(k) X suc tat Let r G r (x ) (V iab G r(k) X ) = ten x / V iab G (K). Proof Let x V iab G (K) X. From definition of te viability kernel, we ave G(x ) V iab G (K) From definition of α(), (G(x ) V iab G (K) + α()b) X ) 14

15 tis implies tat (G(x ) + α()b) (V iab G (K) + α()b) X and for any r max(k, 1)α(), (G(x ) + rb) (V iab G (K) + r k B) X Since G is k-lipscitz, we can apply Lemma 2.3, and tenwe obtain: G r (x ) (V iab G r(k) X ). In particular, if we apply tis Proposition for G = Γ we ave: (K) X suc tat Γ α() x V iab α() Γ ten x / V iab Γ (K) and since from (31) V iab F (K) V iab Γ (K) x / V iab F (K) (x ) (V iab α() Γ (K) X ) =, We can deduce te following approximation result wen K is a viability domain: Corollary 4.1 Let G : X X a k-lipscitz set-valued map and K a viability domain of G. Ten r max(k, 1)α(), V iab G r (K ) = K. Proof Let X K. x belongs to V iab G (K) X and from Proposition 4.2 G r (x ) (V iab G r(k) X ). Since V iab G (K) V iab G r(k) = K, we replace V iab G r(k) by K and ten we obtain: G r (x ) K ) tat is to say tat K is a viability domain of G r. However, we prove tat wen goes to 0, if goes to 0 slower tan a(), we can approximate te viability kernel of K under F by a sequence of finite viability kernels of reduction to X of some larger extensions of 1 + F. We look now for extension G r of G suc tat any solution ξ S G (ξ 0 ) can be approaced by solution ξ S G r (ξ 0 ) 15

16 Lemma 4.1 Let G : X X a k-lipscitz set-valued map. Let r kα(). Let G r : X X te extension of G: x X, G r (x) := G(x) + rb. and consider G r : X X te reduction of G r to X : If te following property olds true: G r (x ) := G r (x )) X, x X. (45) ξ G(x), ξ G(x) X suc tat ξ ξ r k Ten wit any solution ξ := (ξ n ) n S G (ξ 0 ) to te discrete dynamical system: (46) ξ n+1 G(ξ n ), n 0 we can associate a solution ξ := (ξ n) n S G r (ξ 0 ) to te finite dynamical system: (47) ξ n+1 G r (ξ), n n 0 suc tat (48) ξ n ξ n r, n 0. k Proof Let ξ 0 X and ξ S G (ξ 0 ). From definition of α(), since r kα(), ξ 0 ({ξ0 } + r k B) X. Assume tat we found a sequence ξ k satisfying (47) and (48) until k = n. Since G is k-lipscitz, and ten, from (48), G(ξ n ) G(ξ n ) + k ξ n ξ n B G(ξ n ) G r (ξ n ) Since ξ n+1 G(ξ n ), from (45), tere exists ξ n+1 On te oter and we ave ξ n+1 ξ n+1 ξ n+1 r k G(ξ n ) + k ξ n ξ n B G r (ξ n ). Since G(ξ n ) X G r (ξn ), n 0, ten ξ S G r (ξ 0 ). Tis ends te proof of Lemma 4.1. G(ξ n ) X suc tat We deduce te following result: 16

17 x 2 O x 1 A solution x( ) S F (x 0 ) and G = Γ ξ n = x(n), ξ n+1 G(ξ n ) ξ n+1 G r (ξn ) (ξn + r k B) ξ n + r k B G r (ξn ) Figure 1: Te Extension-Reduction Process Corollary 4.2 Let G : X X a k-lipscitz set-valued map satisfying property (45). Let K a closed subset of X Ten, for all r kα() we ave: V iab G (K) V iab G r (K r k ) + r k B Proof From Lemma 4.1, for all ξ 0 V iab G (K), tere exists ξ S G (ξ 0 ) viable in K, ξ 0 K r k and ξ S G r (ξ 0) viable in (K r k )). By definition of te discrete viability kernel, ξ 0 V iab G r (K r k ) and since ξ 0 ξ 0 r k, we ave V iab G (K) V iab G r (K r k ) + r k B Te reduction process satisfies te following property: 17

18 Lemma 4.2 For any closed subset D X, and any decreasing sequence of closed subsets D suc tat D = >0 D, we ave: (49) D = lim sup, 0 ((D + α()b) X ) If D is satisfies te property: x D, x D X (50) D = lim sup 0 (D X ) : x x α(), ten Proof Proof of second statement is immediate. We just prove te first equality. Let x lim sup, 0 (D + α()b) X. Tere exists n, n converging to zero and x n n (D n + α( n )B) X n D n + α( n )B wic converges to x 0 (D + α()b) = D. wen n converge to. Ten x 0 Conversely, let x D. Since D = 0 D, tere exists a sequence x wic converges to x. From definition of α(), tere exists y (D + α()b) X suc tat y x α() and lim, 0 y = x. 4.4 Approximation of V iab F (K) by Viability Kernel of finite subsets V iab Γ r (K r ) Now we can state te following result: Teorem 4.1 Let F : X X a Marcaud and l-lipscitz set-valued map, K a closed subset of Dom(F ) satisfying te boundedness condition: (51) M := sup sup x K y F (x) y < (52) Let G := 1 + F, Γ := 1 + F + Ml 2 2 B and we note k = 1 + l. Let > 0, X a reduction of X and α() defined by (33). Assume tat and are coosen suc tat: Let Γ kml2 α() Ml 2 2 : X X and Γ kml2 : X X defined as follows : Γ kml2 (x) := Γ (x) + kml 2 B Ten: (53) and (54) Γ kml2 (x ) := Γ kml2 (x ) X V iab F (K) = lim sup(v iab Γ (K) + α()b) X, 0 V iab F (K) = lim sup, 0 V iab Γ kml 2 (K Ml2 ) 18

19 Proof From Teorem 3.2 V iab F (K) = lim sup V iab Γ (K) 0 Te sequence of embeded subsets V iab Γ (K) converges to V iab F (K) wen decreases to zero. Ten applying Lemma 4.2, we obtain te first equality (53): V iab F (K) = lim sup(v iab Γ (K) + α()b) X, 0 To prove te second equality (54), we apply Corollary 4.2 wit G = Γ. We ave to ceck first tat assumption (52) implies te tickness condition (45) of Corollary 4.2: indeed ξ Γ (x), ξ x + F (x) suc tat ξ ξ Ml 2 2 From te definition of α() Since from (52), (x + F (x) + α()b) X ξ X suc tat ξ ξ α() ( x + F (x) + Ml ) 2 2 B X = Γ (x) X Ten we proved tat ξ Γ (x), ξ Γ (x) X suc tat ξ ξ ξ ξ + ξ ξ Ml 2 We are able now to apply Corollary 4.2 wit r = kml 2 and tus we obtain: wic implies tat and ten (55) V iab Γ (K) V iab Γ kml 2 V iab F (K) lim sup, 0 ( V iab F (K) lim sup, 0 V iab Γ kml 2 (K Ml2 ) + Ml 2 B ) (K Ml2 ) + Ml 2 B V iab Γ kml 2 (K Ml2 ) To prove te opposite inclusion, we observe tat and ten we ave Grap Γ kml2 = 1 + F + 2 ΓkMl ( Ml kml 2 ) B 1 Grap(F ) + ( l)mlb 19

20 and if we assume for instance tat l 1 2, Ten, and ( 3 + l)ml 2Ml 2 ɛ > 0, ɛ > 0 suc tat ]0, ɛ [ : 2Ml ɛ ɛ, > 0 suc tat ]0, ɛ, [, : α() Ml 2 2 Te Convergence Teorem 3.1 implies tat: lim sup, 0 V iab Γ kml 2 and wit (55) we proved te equality lim sup, 0 V iab Γ kml 2 (K Ml2 ) V iab F (K). (K Ml2 ) = V iab F (K) 4.5 Conclusion : a numerical metod for computing viability kernel Tese results allow us to look for numerical approximation of te viability kernel of K under F associated wit te initial differential inclusion (1): ẋ(t) F (x(t)), for almost all t 0. We consider te discrete explicit sceme: { x n+1 x n + F (x n ) + 2Ml 2 B, n 0, x 0 = x 0 K, We recall tat te condition (56) (x + F (x ) + rb X will be true if and satisfy te condition: (57) r = 2Ml 2 α() witc is a stability condition meaning tat te space discretization step as to be smaller tan te time s one. We set Γ (x) := x + F (x) + Ml 2 2 B 20

21 G 2Ml2 (x) := x + F (x) + 2Ml 2 B From (13) in te discrete case and (37) in te finite case we obtain: but K Ml 2 2, can be empty, and K Ml 2 2, := V iab Γ (K). K Ml 2 2, := V iab Γ (K ). K 2Ml2, := V iab G 2Ml 2 (K). K 2Ml2, := V iab G 2Ml 2 (K Ml2 ). but now, if > 0 and > 0 satisfy te condition (57), te finite viability kernel is non empty. Gatering general results we proved in preceeding sections, we ave te following convergence properties of approximations of viability kernel of K under F wit finite viability kernels computable in a finite number of steps: K 2Ml2, Teorem 4.2 If F is a Marcaud setvalued map, K a compact subset of X, > 0 and > 0 satisfying te condition (57). Ten K Ml 2 2,, K Ml 2 2, X lim 0 K Ml 2 2, = K lim sup 0 K V iab F (K) Moreover, if F is l-lipscitz, ten References lim sup K 2Ml2,, = V iab F (K), 0 [1] AUBIN J.-P. & CELINA A. (1984) Differential Inclusions Springer-Verlag, Berlin [2] AUBIN J.-P.& FRANKOWSKA H. (1990) Set-valued analysis. Birkaüser. [3] AUBIN J.-P. Viability Teory. To appear. [4] BERGE C. (1966) Espaces Topologiques, Fonctions Multivoques Dunod, Paris [5] BYRNES C.I. & ISIDORI A. (1990) Régulation Asymptotique de Systèmes non Linéaires C.R.A.S., Paris, 309,

22 [6] FILIPPOV A.F. (1967) Classical solutions of differential equations wit multivalued rigt and side. SIAM, J. on Control, 5, [7] FRANKOWSKA H. (To appear) Set-valued Analysis and Control Teory. Birkaüser. [8] FRANKOWSKA H. & QUINCAMPOIX M. (1991) Fast Viability Kernel Algoritm. [9] HADDAD G. (1981) Monotone viable Trajectories for Functional Differential Inclusions. J.Diff.Eq., 42, 1-24 [10] QUINCAMPOIX M. (1990) Frontière de domaines d invariance et de viabilité pour des inclusions différentielles avec contraintes. Comptes Rendus Académie des Sciences, Paris [11] SAINT-PIERRE P. (1990) Approximation of Slow Solutions to Differential Inclusions. Applied Matematics and Optimisation. 22, [12] SAINT-PIERRE P. (1991) Viability of Boundary of te Viability kernel. To appear 22