Filippov's operator and discontinuous differential equations

Abstract

The thesis is mainly concerned about properties of the so-called Filippov operator that is associated with a differential inclusion x'(t) ε F(x(t)) a.e. t ε [0,T], where F : Rn → Rn is given set-valued map. The operator F produces a new set-valued map F[F], which in effect regularizes F so that F[F] has nicer properties. After presenting its definition, we show that F[F] is always upper-semicontinuous as a map from Rn to the metric space of compact subsets of Rn endowed with the Hausdorff metric. Our main approach is to study the operator via its support function, which we show is an upper semicontinuous function. We show that the support function can be used to characterize the operator, and prove a new result that characterizes those set-valued maps that are fixed by F; this result was previously known to hold only in dimension one. We also generalize to higher dimensions a known result that characterizes those set-valued maps that are almost everywhere singleton-valued (that is, F(x) = {f(x)} where f : Rn → Rn is an ordinary function). The latter part of the thesis introduces four generalized solution concepts of discontinuous differential equations. These are known as the Filippov, Krasovskij, Hermes, and Euler solution concepts. We study the relations among these solution concepts, and in particular prove that the Euler and Hermes solutions in the autonomous case coincide.