Upper and lower semicontinuous differential inclusions: A unified approach.

*(English)*Zbl 0704.49011
Nonlinear controllability and optimal control, Lect. Workshop, New Brunswick/NJ (USA) 1987, Pure Appl. Math., Marcel Dekker 133, 21-31 (1990).

[For the entire collection see Zbl 0699.00040.]

The basic result states that, given an open set \(A\subseteq {\mathbb{R}}\times {\mathbb{R}}^ n\) and a bounded lower semicontinuous multifunction G: \(A\mapsto {\mathbb{R}}^ n\) with closed (not necessarily convex) values, there exists an upper semicontinuous map F: \(A\mapsto {\mathbb{R}}^ n\) with compact convex values such that every solution of \(\dot x\in F(t,x)\) is also a solution of \(\dot x\in G(t,x)\). To obtain F, one should first construct a directionally continuous selection f(t,x)\(\in G(t,x)\), then define F as the upper semicontinuous, convex valued regularization of f.

Using the above theorem, several statements concerning lower semicontinuous differential inclusions can be easily deduced from the corresponding results, known in the upper semicontinuous, convex-valued case. In particular, theorems concerning the existence of solutions on closed sets, the existence of periodic solutions, and the connectedness of the reachable set are proved.

The basic result states that, given an open set \(A\subseteq {\mathbb{R}}\times {\mathbb{R}}^ n\) and a bounded lower semicontinuous multifunction G: \(A\mapsto {\mathbb{R}}^ n\) with closed (not necessarily convex) values, there exists an upper semicontinuous map F: \(A\mapsto {\mathbb{R}}^ n\) with compact convex values such that every solution of \(\dot x\in F(t,x)\) is also a solution of \(\dot x\in G(t,x)\). To obtain F, one should first construct a directionally continuous selection f(t,x)\(\in G(t,x)\), then define F as the upper semicontinuous, convex valued regularization of f.

Using the above theorem, several statements concerning lower semicontinuous differential inclusions can be easily deduced from the corresponding results, known in the upper semicontinuous, convex-valued case. In particular, theorems concerning the existence of solutions on closed sets, the existence of periodic solutions, and the connectedness of the reachable set are proved.

Reviewer: Alberto Bressan