Criteria for viability of trajectories of nonautonomous differential inclusions and their applications (Q1323203)

The author gives a necessary and sufficient condition for the existence of (viable) solutions of the problem \([\dot x(t) \in F(t,x(t))\), \(x(\tau) = x\), \(x(t) \in W(t)\), \(t \geq \tau]\), where the map \(F(t,\cdot)\) is upper semicontinuous, \(F(\cdot,x)\) is measurable, \(\| F(t,x) \| \leq r\) for some \(r>0\), and the graph of \(W\) is closed. This condition is expressed in terms of the contingent derivative of the map \(G_{\tau,x} (t) = W(t) - X(t,\tau,x)\), where \(X (t, \tau,x)\) 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.