Convex continuation of the convex set valued maps and applications to the differential inclusion theory

ABSTRACT PhD Thesis CONVEX CONTINUATION OF THE CONVEX SET VALUED MAPS AND APPLICATIONS TO THE DIFFERENTIAL INCLUSION THEORY SERKAN ALI DUZCE Anadolu University Graduate School of Natural And Applied Sciences Mathematics Program Supervisor: Assoc. Prof. Kh.G. GUSEINOV 2003, 124 Pages In this thesis, convex continuation for the compact, convex set valued maps on an interval was defined, existence of the convex continuation of the compact, convex set valued maps was studied and the cases which there is no convex continuation were investigated. Maximality of the convex continuation was also considered. The relation between the existence of the convex continuation of the compact, convex set valued maps and the Lipschitz continuity of the set valued maps was investigated. Finally, an inverse problem of the differential inclusion theory was considered. It is required to define a differential inclusion so that the Hausdorff distance between the reachable sets of this inclusion and the value sets of the given set valued map does not exceed the given value. As a first step, the problem was solved for a special case where the set valued map is an affine tube. Then, on generalizing this approach, a solution was obtained for the compact, convex valued, continuous set valued maps. Keywords: Convex Continuation, Set Valued Map, Differential Inclusion, Reachable Set, Integral Funnel. u