Near Viability of a Set-Valued Map Graph with Respect to a Quasi-Autonomous Nonlinear Inclusion

Abstract

This paper addresses near viability of a set-valued map graph G with respect to a quasiautonomous fully nonlinear differential inclusion of the form y

(t) ∈ Ay(t )+F(t, y(t)). We introduce a new notion of A-quasi-tangency when A is a nonlinear m-dissipative set-valued operator.We give necessary and sufficient conditions for G to be near viable with respect to the previous differential inclusion. We obtain under weak hypotheses a classical relaxation result stating that each solution of the relaxed differential inclusion can be approximated by a solution of the differential inclusion at any given precision