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On the second-order contingent set and differential inclusions. (English) Zbl 0958.34010

The authors establish the existence of solutions to a nonconvex second-order differential inclusion of the following type: \[ \ddot{x}(t)\in F(x(t), \dot{x}(t)) \text{ a.e, } x(0)=x_0\in K, \dot{x}(0)=v_0\in \Omega, \] such that \(x(t)\in K\), where \(K\) is a closed subset and \(\Omega\) is an open subset of \(\mathbb{R}^n\). When \(K\) is in addition convex, they introduce the contingent cone \(T_K\) to prove the existence of solutions to the differential inclusion: \[ \ddot{x}(t)\in G(x(t),\dot{x}(t)) \text{ a.e, } x(t)\in K \text{ and } \dot{x}(t)\in T_K(x(t)). \]

MSC:

34A60 Ordinary differential inclusions
49J24 Optimal control problems with differential inclusions (existence) (MSC2000)
34H05 Control problems involving ordinary differential equations
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