Approximation of the viability kernel. (English) Zbl 0790.65081

Saint-Pierre, Patrick

Appl. Math. Optimization 29, No. 2, 187-209 (1994).

Let \(X\) be a finite-dimensional vector space, \(K\) a compact subset of \(X\). In the paper the viability kernel \(V\) (\(V_ \rho\)) of \(K\) under \(F\) (\(G_ \rho\)) of the differential inclusion \(\dot x(t) \in F(x(t))\), \(x(0) = x_ 0 \in K\) (of its discrete explicit scheme \(x^{n+1} \in G_ \rho(x^ n)\), \(n \geq 0\), \(G_ \rho = 1 + \rho F\), i.e. \((x^{n+1} - x^ n)/\rho \in F(x^ n))\) is studied where \(F\) is a Marchaud map from \(K\) to \(X\). It is proved that, for a good choice of discretizations, the sequence of \(V_ \rho\) converges to \(V\) if \(F\) is Lipschitzian.

Reviewer: M.Bartušek (Brno)

MSC:

65L99	Numerical methods for ordinary differential equations
34A60	Ordinary differential inclusions

Keywords:

convergence; viability kernel; differential inclusion; Marchaud map; choice of discretizations

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