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Zbl 0798.34019
Ledyaev, Yu.S.
Criteria for viability of trajectories of nonautonomous differential inclusions and their applications.
(English)
[J] J. Math. Anal. Appl. 182, No.1, 165-188 (1994). ISSN 0022-247X

The author gives a necessary and sufficient condition for the existence of (viable) solutions of the problem $[\dot x(t) \in F(t,x(t))$, $x(\tau) = x$, $x(t) \in W(t)$, $t \ge \tau]$, where the map $F(t,\cdot)$ is upper semicontinuous, $F(\cdot,x)$ is measurable, $\Vert F(t,x) \Vert \le r$ for some $r>0$, and the graph of $W$ is closed. This condition is expressed in terms of the contingent derivative of the map $G\sb{\tau,x} (t) = W(t) - X(t,\tau,x)$, 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.
[A.L.Dontchev (Ann Arbor)]
MSC 2000:
*34A60 ODE with multivalued right-hand sides

Keywords: contingent derivative; differential inclusion; comparison of solutions; viable trajectories

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