On the set of solutions to Lipschitzian differential inclusions. (English) Zbl 0723.34009

Cellina, Arrigo

Differ. Integral Equ. 1, No. 4, 495-500 (1988).

We consider a Lipschitzian differential inclusion \((1)\quad x'(t)\in F(t,x),\quad x(t_ 0)=a,\) where the values of F are compact but not convex. We prove that the map that associates to the initial point a the set of solutions to (1), \(S_ F(a)\), admits a selection, continuous from \(R^ n\) to the space of absolutely continuous functions. The images of this map are sets that are not decomposable. In particular, the map from a to the attainable set at T, \(A_ T(a)\), admits a continuous selection. It is known that this map, in general, has no closed values.

MSC:

34A60	Ordinary differential inclusions
49J45	Methods involving semicontinuity and convergence; relaxation
93B03	Attainable sets, reachability

Keywords:

Lipschitzian differential inclusion; compact but not convex; continuous selection