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Criteria for viability of trajectories of nonautonomous differential inclusions and their applications. (English) Zbl 0798.34019
The author gives a necessary and sufficient condition for the existence of (viable) solutions of the problem [x ˙(t)F(t,x(t)), x(τ)=x, x(t)W(t), tτ], where the map F(t,·) is upper semicontinuous, F(·,x) is measurable, F(t,x)r for some r>0, and the graph of W is closed. This condition is expressed in terms of the contingent derivative of the map G τ,x (t)=W(t)-X(t,τ,x), where X(t,τ,x) is the reachable set of the differential inclusion. It is used for comparison of solutions to differential equations and generalized differential inequalities. Under some additional assumptions a differential inclusion whose trajectories coincide with viable trajectories of the original problem is constructed.
34A60Differential inclusions