Second order viability problems for differential inclusions (Q1323235)

The paper concerns a ``viability'' problem for the second order differential inclusion \(x''(t) \in F(x(t)\), \(x'(t))\) where \(F:G(T_ L) \to\mathbb{R}^ n\) is a set-valued map \((G\) stands for graph, \(T\) stands for contingent cone) and \(L \subseteq\mathbb{R}^ n\) is a set. Given another set \(M \subseteq\mathbb{R}^ n\) and a solution \(x : [0,T[ \to\mathbb{R}^ n\) to the differential inclusion above, with \(x(0) \in M\), the problem is whether there exists \(T_ 0 \in]0,T]\) such that \(x(t) \in M\) for all \(t \in [0, T_ 0[\). A necessary condition and a sufficient condition are established by using a set of second order tangents of \(L\) at \((x(0),x'(0))\) and a set of second order interior tangents of \(M\) at the same point. The arguments revolve around the set of limit points of \((x(t) - x(0) - tx' (0))/(t^ 2/2)\) as \(t \to 0 +\) (cf. \textit{T. Wazewski} [Bull. Acad. Polon. Sci., Ser. Sci. Math. Astron. Phys. 9, 865- 867 (1961; Zbl 0101.060)]).