ON CONDITIONS FOR CONTROLLABILITY AND LOCAL REGULARITY OF A SYSTEM OF DIFFERENTIAL EQUATIONS

• Main Author: Hadra Zuhri, Ahmad
• Format: Theses
• Language: Indonesia
• Online Access: https://digilib.itb.ac.id/gdl/view/71964
• Institution: Institut Teknologi Bandung

Consider a system of differential equation on a Banach space ???? given by: ?????(????)=????????(????)+????(????)????(????,????(????)),????(0)=????0, where ???? is an infinitesimal generator of an ????-dissipative operator, ????:?0+×????????? is a function with certain Lipschitz property, and ?????????????([0,????],?) is a control defined on [0,????] with 1<????<?. One of the physical system satisfied by the above system is heat equation problem with an external source ????, and an infinitesimal generator in the form of Laplacian. In this thesis, the controllability of the system for a ????-bounded function ????, and the local regularity of the system for a locally bounded function ???? are examined. Here, the controllability of the system refers to the ability of the control to carry a system from the arbitrary initial state to the desired final states. In particular, the above system is said to be controllable if its reachable set or the set of all the final states of the system is compact. Using Gronwall’s Lemma, the equivalence of the mild solution of the system for ???? that is Lipschitz on bounded sets on ???? and ????-bounded and ????? that is globally Lipschitz is shown. A straightforward application of the corollary of Compactness Principle Theorem yields the compactness and locally Lipschitz property of the reachable set of the system. Hence, the system is controllable for a ????-bounded function ????. The local regularity property in this thesis refers to the continuity of the solution of the system. The local existence and the uniqueness of the solution of the system are established. The main tools that used in proving local regularity are ????-weighted norms and Banach Fixed Point Theorem.