Type of Document Dissertation Author Alshammari, Khalid Abdulaziz Author's Email Address kalsha1@lsu.edu URN etd-04042006-174834 Title Filippov's Operator and Discontinuous Differential Equations Degree Doctor of Philosophy (Ph.D.) Department Mathematics Advisory Committee

Advisor Name Title Peter Wolenski Committee Chair Frank Neubrander Committee Member Guillermo Ferreyra Committee Member Robert F. Lax Committee Member Robert Perlis Committee Member Morteza Naraghi-Pour Dean's Representative Keywords

- filippov operator
- krasovskij's solution
- filippov's solution
- euler's solution
- heremes's solution
- discontinuous differential equations
Date of Defense 2006-03-14 Availability unrestricted AbstractThe 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 : R^{n}→ R^{n}is given set-valued map. The operatorFproduces a new set-valued mapF[F], which in effect regularizes F so thatF[F] has nicer properties. After presenting its definition, we show thatF[F] is always upper-semicontinuous as a map from R^{n}to the metric space of compact subsets of R^{n}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 byF; 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 : R^{n}→ R^{n}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.

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