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A measurable upper semicontinuous viability theorem for tubes. (English) Zbl 0838.34017

A viability problem \(x'\in F(t, x)\), \(x(t)\in P(t)\) is considered. It is assumed that \(F\) is jointly measurable, upper semicontinuous in \(x\) with nonempty closed convex (not necessarily bounded) values and \(P\) is absolutely continuous. A necessary and sufficient condition (expressed through the standard tangential condition) for the existence of a viable solution to the above problem is given. Using Lyapunov functions, monotone solutions of the above problem are studied. The existence of a smallest lower semicontinuous Lyapunov function is proved.

MSC:

34A60 Ordinary differential inclusions
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
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