One-sided Perron Differential Inclusions

The main qualitative properties of the solution set of almost lower (upper) semicontinuous one-sided Perron differential inclusion with state constraints in finite dimensional spaces are studied. Using the technique introduced by Veliov (Nonlinear Anal 23:1027–1038, 1994) we give sufficient conditions for the solution map of the above state constrained differential inclusion to be continuous in the sense of Hausdorff metric. An application on the propagation of the continuity of the state constrained minimum time function associated with the nonautonomous differential inclusion and the target zero is given. Some relaxation theorems are proved, which are used afterward to derive necessary and sufficient conditions for invariance.