Approximation of the Viability Kernel

We study recursive inclusions x n +1 ∈ G ( x n ). For instance such systems appear for discrete ﬁnite diﬀerence inclusions x n +1 ∈ G ρ ( x n ) where G ρ := 1 + ρF . The discrete viability kernel of G ρ , i.e. the largest discrete viability domain, can be an internal approximation of the viability kernel of K under F . We study discrete and ﬁnite dynamical systems. In the Lipschitz case we get a generalization to diﬀerential inclusions of Euler and Runge-Kutta methods. We prove ﬁrst that the viability kernel of K under F can be approached by a sequence of discrete viability kernels :associated with Γ ρ ( x ) = x + ρF ( x ) + Ml 2 ρ 2 B . Secondly, we show that it can be approached by ﬁnite viability kernels associated with Γ αhρ ( x ) := x + ρF ( x ) : x n +1 h ∈ (Γ hρ ( x nh ) + α ( h ) B ) ∩ X h .


Introduction
Let X a finite dimentional vector space and K a compact subset of X.
Let us consider the differential inclusion: ẋ(t) ∈ F (x(t)), for almost all t ≥ 0, where F is a Marchaud map 1 defined from K to X.
With this inclusion, for a fixed ρ > 0, we associate the discrete explicit scheme: We note G ρ the set-valued map G ρ = 1 + ρF and the system (2) can be rewrited as follows: x n+1 ∈ G ρ (x n ), for all n ≥ 0, (3) F is upper-semicontinuous, convex compact valued ∀x ∈ Dom(F ), F (x) := max y∈F (x) y ≤ c( x + 1) The Viability Theory allows to study viable solutions of (1) and the subset of elements x 0 ∈ K such that there exists at least a viable solution starting at x 0 .On the other hand, we look for approximation of such solutions and we wonder how the set of initial points from which there exists at least a viable approximation solution to (2) and the set of initial points from which there exists at least a viable solution to (1) are related together.
These sets are called viability kernel of K under F or discrete viability kernel of K under Gρ.Byrnes & Isidori [5] and Frankowska & Quincampoix [8] have proposed algorithms which approximate the viability kernel of K under F when F is lipschitzian and K is closed.
We prove that, when F is a Marchaud map, for a good choice of discretizations Gρ, the sequence of discrete viability kernels of K under Gρ converges to a subset contained in the viability kernel of K under F .Moreover it converges to the viability kernel if F is lipschitzian.
We show that similar results remain true when we introduce a discretization of the space and consider finite viability kernels.

Definitions and General Results
We call discrete dynamical system associated with G the following system: We denote by -K the 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 ) the set of solutions x ∈ X to the discrete dynamical system starting at x 0 .
A solution x is viable if and only if x ∈ S G (x) ∩ K: It means that there exists a selection of equation ( 2) which remains in K at each step n.
We study the subset of initial points in K from which there exists at least one viable solution.
Let K be a subset of X.The discrete viability kernel of K under G is the largest closed discrete viability domain contained in K and we denote it V iab G (K).
We can point out the following remark and properties: is the 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 the discrete viability kernel V iab G (K) while it remains in K.
Moreover any solution of (5) which does not start from V iab G (K) must leave K in a finite number of steps.

A Construction Method for Discrete Viability Kernel
Let us consider the sequence of subsets K 0 = K, K 1 , ..., K n , ... defined as follows: We note Proposition 2.1 Let G: X X a upper semicontinuous set-valued map with closed values and K a compact subset of Dom(G).Then is closed and, for all x ∈ K 0 \K 1 , it does not exit any viable solution starting from x.This implies recursively that Definition 2.2 Let G : X X a set-valued map and r > 0. We call extension of G with a ball of radius r the set-valued map G r : X X defined by : We consider the sequence of subsets K r,0 = K, K r,1 , ..., K r,n , ... defined as follows: If G is an upper semicontinuous set-valued map, G r : X X is also upper semicontinuous and from Proposition 2.1: When r decreases to 0, the viability kernel of K under G r converges to the viability kernel of K under G: Proposition 2.2 Let G be upper semicontinuous and K a compact subset of X.The following property holds: is compact and both are, from (8), decreasing sets when r decreases to zero.Also the intersection G r (x 0 ) ∩ V iab G r (K) is a decreasing sequence of nonempty compact sets and When G is a k-Lipschitz setvalued map, we have the following result giving an estimation of the growth of the discrete viability kernel when r increases: Then: From (16), we deduce that there exists a viable solution for the system associated with G r starting from x and thus x ∈ V iab G r (K).

Approximation of Viability Kernels for Finite Difference Inclusions
Let F a Marchaud map and Γ ρ a sequence of setvalued maps which correspond to discretizations associated with the initial differential inclusion (1) satisfying: where B is the unit ball in X × X.
We note Assumption (17) implies that the graph of F contains the graphical upper limit 3 of F ρ , that is to say that Graph(F ) contains the Painlevé-Kuratowski upper limit 4 of Graph(F ρ ): .Then the upper limit K = lim sup ρ→0 V iab Γρ (K ρ ) is a viable subset under F :
From the definition of Γ ρ , x n+1 ρ ∈ Γ ρ (x n ρ ) and then: 3 The graphical upper limit is the upper limit of the sequence of Graph(Fρ). 4The upper limit of a sequence of subsets Dn of X is With this sequence we associate the piecewise linear interpolation x ρ (•) which coincides to x n ρ at nodes nρ: Since F is Marchaud, and from (18), set-valued maps F ρ satisfy a uniform linear growth: As in the proof of the Viability Theorem (see [2], [9]), this implies and with (17) we have By the Ascoli and Alaoglu Theorems, we derive that there exists x(•) ∈ W 1,1 (0, +∞; X; e −ct dt) and a subsequence (again denoted by) x ρ which satisfy: This implies (see [1] The Convergence Theorem) that x(•) is a solution to the differential inclusion: It remains to prove that the limit is a viable solution: ∀t > 0, there exists a sequence n t = E( t ρ ) such that n t ρ → t when ρ → 0. Then x(t) = lim ρ→0 x ρ (n t ρ).Since ∀ρ:

Examples of Approximation Processes
1 -The finite difference explicit scheme.
Naturally, the discrete explicit scheme (2) Let us define the set-valued Runge-Kutta scheme Γ RK ρ : For any x ∈ K, Then condition (17) holds and from Theorem 3.1 we deduce the following corollary:

-The thickening process.
Let us define the set-valued map F T ρ : X → X by a thickening of the values of F by balls of radius M l 2 ρ: and consider the set-valued map associated with the finite differnece scheme for When F is Marchaud, we have the following relations between V iab Gρ (K), V iab Γ T ρ (K) and V iab F (K) : Corollary 3.2 Let F a Marchaud map, G ρ and Γ ρ defined by ( 22).Then Proof --The first inclusion holds true since G ρ (x) ⊂ Γ T ρ (x) and ( 8).On the other hand, since Theorem 3.1 implies the second inclusion.

Approximation of the Viability Kernel in the Lipschitz case
From now on we use the following notations : When F is l-Lipschitz, we claim that the discrete viability kernel V iab Γρ (K) is a good approximation of the viability kernel of K under F .Theorem 3.2 5 Let F a Marchaud and l-Lipschitz set-valued map, let K a closed subset of X satisfying the boundedness condition We want to check the opposite inclusion.
Let x 0 ∈ K and consider any solution x(•) ∈ S F (x 0 ).Let ρ > 0 given.We have So, we have proved that if x(•) ∈ S F (x 0 ) then the following sequence is a solution to the discrete dynamical system associated with Γ ρ :

Approximation by Finite Setvalued Maps
With any h ∈ IR we associate X h a countable subset of X, which spans X in the sense that where α(h) decreases to 0 when h → 0:

Approximation of discrete and finite viability kernels
Let G h : X h X h a finite set-valued map and a subset K h ⊂ Dom(G h ).We call finite dynamical system associated with G h the following system: and we denote by -K h the set of all sequences from IN to K h .
-x h := (x 0 h , ..., x n h , ...) ∈ X h a solution to system (35) -S G h (x 0 h ) the set of solutions x h ∈ X h to the finite differential inclusion (35) starting from x 0 h A solution x h is viable if and only if x h ∈ S G h (x h ) ∩ K h , that is to say that: .. defined recursively as in the second section: The viability kernel algorithm and Proposition 2.1 holds true for finite dynamical systems whenever the set-valued map G h has nonempty values and we have: Let us notice that K ∞ h can be emptyset and in any case there exists a finite integer p such that: We cannot apply no longer more Proposition 2.1 since G(x h ) may not contain any point of the reduction X h of X and G h (x h ) be empty.
To turnover this difficulty, we will consider greater set-valued maps G r which still approximate G.But the choice of such approximations is subjet to two opposite considerations: on one hand, they have to be large enough in order that the reductions to X h of such approximations have their domain containing K h (have nonempty values on K h ), and so, it will be possible to apply again Proposition 2.1.On the other hand, the enlargement is limited as far as the graphical assumption (1) of Theorem 3.1 still holds so as to the viability kernel of K under G contains the upper limit of finite viability kernels of K h under the finite set-valued approximations G r h .In case of upper semicontinuous set-valued maps, we bring in the fore some discretization process which leads to approach a subset of a the viability kernel.
In the Lipschitz case, these process enables us to approach the viability kernel completely.

Notations:
the reduction to the finite subset X h of any subset D will be noted by a lower index h: D h := D ∩ X h ; the extension of a set-valued map G with a ball of radius r by an upper index: ∀x ∈ X, G r (x) := G(x) + rB.Thus the reduction to X h of the extension of a set-valued map G with a ball of radius r will be noted G r h .Let us notice that the extension operation has to be done before the reduction one otherwise it could be empty even for r > α(h).
From property (33) which defines α(h), we consider now the extension with r = α(h).We observe that G α(h) h satisfies the non emptyness property: and the decreasing sequence of finite subsets K As a partial conclusion, we are able to approximate the discrete viability kernel of K under G: first we extend G such that for all x ∈ K, images of G r encounters X h , in other words such that Dom(G r h ) = Dom(G) ∩ X h .To be sure of this, without loss of generality, we can choose r = α(h).Secondly we look after the discrete viability kernel of K under G r , the finite viability kernel of K h under G r h and at last we let h decreasing to 0. What relations link together the discrete viability kernels whenever K h is the reduction of K to X h : are the latters the reduction to X h of the formers ?Does the upper limit of the latters, when h goes to 0, coincide with the former ?

Properties of the finite viability kernel
A first answer is given by applying Proposition 2.2: since lim h→0 α(h) = 0, we have The following result gives a necessary and sufficient condition for ) defined as follows: Let Proof -From (37) we have to check that the two following statements Assume that (42) holds.Let and we obtain inclusion: Then from definition 2.1 and definition of G r h it is also a finite viability domain of G r h contained in K h and thus is contained in the finite viability kernel of We obtain the opposite inclusion and prove that (43) is true.
Conversely, if (43) holds, V iab G r (K) ∩ X h is the finite viability kernel of K h under G r h , it is obviously a finite viability domain of G r . Remarks 2 -It is easy to prove that B ⊂ A and Proposition 4.1 says that if B is empty, A is empty too.
3 -If A = ∅, then all solutions x h to the finite dynamical system starting from any point x 0 h ∈ A, must leaves K h after a finite number of steps, although x 0 belongs to the dicrete viability kernel of K under G r .4 -If B = ∅, then all solutions x h to the finite dynamical system starting from any point x 0 h ∈ B, leaves K h at the first step.

-If x 0
h ∈ A and x n h is the last element of a solution to the finite dynamical system, starting at x 0 h , which is still in K h , then Then inclusion (40) becomes an equality :

Approximation of the viability kernel of K under F by finite viability kernel in the Lipschitz Case
When G is a k-Lipschitz set-valued map, we cannot prove that in (40), the inclusion becomes an equality.Nevertheless we have in the Lipschitz case an immediate and interesting information about points x h ∈ K h which do not satisfy (41): From definition of the viability kernel, we have and for any r ≥ max(k, 1)α(h), Since G is k-Lipschitz, we can apply Lemma 2.3, and thenwe obtain: In particular, if we apply this Proposition for G = Γ ρ we have: and since from (31 We can deduce the following approximation result when K is a viability domain: Corollary 4.1 Let G : X → X a k-Lipschitz set-valued map and K a viability domain of G. Then K) by K and then we obtain: However, we prove that when h goes to 0, if ρ goes to 0 "slower" than a(h), we can approximate the viability kernel of K under F by a sequence of finite viability kernels of reduction to X h of some larger extensions of 1 + ρF .
We look now for extension G r of G such that any solution ξ ∈ S G (ξ 0 ) can be approached by solution If the following property holds true: Then with any solution ξ := (ξ n ) n ∈ S G (ξ 0 ) to the discrete dynamical system: we can associate a solution ξ h := (ξ n h ) n ∈ S G r h (ξ 0 h ) to the finite dynamical system: Proof -Let ξ 0 ∈ X and ξ ∈ S G (ξ 0 ).From definition of α(h), since r ≥ kα(h), ∃ξ 0 h ∈ ({ξ 0 } + r k B) ∩ X h .Assume that we found a sequence ξ k h satisfying (47) and (48 On the other hand we have ).This ends the proof of Lemma 4.1.
We deduce the following result: The Extension-Reduction Process Corollary 4.2 Let G : X X a k-Lipschitz set-valued map satisfying property (45).Let K a closed subset of X Then, for all r ≥ kα(h) we have: The reduction process satisfies the following property: and if we assume for instance that ρl ≤ ) = V iab F (K)

Conclusion : a numerical method for computing viability kernel
These results allow us to look for numerical approximation of the viability kernel of K under F associated with the initial differential inclusion (1): ẋ(t) ∈ F (x(t)), for almost all t ≥ 0.
We consider the discrete explicit scheme: x n+1 ∈ x n + ρF (x n ) + 2M lρ 2 B, ∀n ≥ 0, We recall that the condition (x h + ρF (x h ) + rB ∩ X h = ∅ (56) will be true if ρ and h satisfy the condition: witch is a stability condition meaning that the space discretization step has to be "smaller" than the time's one.From (13) in the discrete case and (37) in the finite case we obtain: can be empty, and but now, if h > 0 and ρ > 0 satisfy the condition (57), the finite viability kernel K 2M lρ 2 ,∞ ρh is non empty.Gathering general results we proved in preceeding sections, we have the following convergence properties of approximations of viability kernel of K under F with finite viability kernels computable in a finite number of steps: Theorem 4.2 If F is a Marchaud setvalued map, K a compact subset of X, h > 0 and ρ > 0 satisfying the condition (57).

Proposition 4 . 1
Let G: X X an upper semicontinuous set-valued map with closed values and K a closed subset of Dom(G).