Topological structure of solution sets of differential inclusions: the constrained case

Wojciech Kryszewski

doi:10.1155/s1085337503204115

Abstract

We survey and announce some current results on the existence, the viability, and the topological structure of the viable solutions of differential equations and inclusion in Banach spaces under set constraints. Some new results concerning semilinear differential inclusions with state variables constrained to the so-called regular and strictly regular sets, together with their applications, are presented and discussed.

Highlights

It is our purpose to study solutions of the Cauchy problem for a semilinear differential inclusion u (t) ∈ Au(t) + φ t, u(t), t ∈ J, u ∈ D, u t0 = x0 ∈ D, (1.1)where φ : J × D E is an upper-Caratheodory set-valued map, J is an interval of the real axis R, t0 ∈ J, D is a closed subset of a Banach space E, and A is the infinitesimal generator of a C0-semigroup {U(t)}t≥0 of bounded linear operators on E (A ≡ 0 and/or E = RN is not excluded)
Where φ : J × D E is an upper-Caratheodory set-valued map, J is an interval of the real axis R, t0 ∈ J, D is a closed subset of a Banach space E, and A is the infinitesimal generator of a C0-semigroup {U(t)}t≥0 of bounded linear operators on E (A ≡ 0 and/or E = RN is not excluded)
326 Structure of solution sets of (1.1) such that u(t) ∈ D on J; we determine the topological structure of the set of such viable solutions

Summary

Introduction

(H3) φ transforms precompact subsets of J × D into compact ones This assumption is automatically satisfied if φ is (jointly) upper semicontinuous or if dim E < ∞ and c ≡ const and seems to be a minimal compactness condition required in an infinite dimensional setting and in the presence of constraints. (H4) A closed densely defined linear operator A : E ⊃ D(A) → E is the infinitesimal generator of a C0-semigroup ᐁ = {U(t)}t≥0 such that U(t) ≤ exp(ωt) where ω ∈ R for t ≥ 0 It is clear (using an appropriate renorming procedure) that this does not restrict generality (for details, cf [50, Chapter VII] and [53, 58]). It is easy to see that S is homeomorphic to the inverse limit lim invk→∞ Sk of the inverse system {Sk, πkl}, where πkl is the restriction of functions from Sk to Jl (l ≤ k)

Existence

Solution sets of the semilinear system

Convex case

Epi-Lipschitz case

Regular case

Final remarks