Approximation of the viability kernel

We study recursive inclusions xn+1 ∈ G(xn). For instance such systems appear for discrete finite difference inclusions xn+1 ∈ Gρ(xn) where Gρ: = 1 + ρF. The discrete viability kernel of Gρ, 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 Γρ(x) = x + ρF (x) + Ml 2 ρ 2B. Secondly, we show that it can be approached by finite viability kernels associated with Γαhρ(x): = x+ ρF (x) : x n+1 h ∈ (Γhρ(xnh) + α(h)B) ∩Xh.