Study of dynamic equations on time scales with applications in control theory (PhD)

Abstract

The present research work deals with the investigation of various kinds of dynamic equations on time scales infinite as well as in infinite-dimensional spaces. This work provides insight into the different types of existence, uniqueness, stability, controllability and observability problems for dynamic equations with impulsive conditions on time scales. We establish the controllability results for time-varying neutral differential equations with impulses on time scales in a finite-dimensional space Rn. Also, we study the controllability results for Volterra integro-dynamic inclusions with impulsive conditions on time scales. Further, we study the controllability and observability results for a dynamic system with noninstantaneous impulses on time scales in a finite-dimensional space Rn by using the variation of parameter and Gramian matrices. Next, we establish existence, uniqueness, stability, and controllability results for Volterra integro-dynamic systems with non-instantaneous impulses on time scales by introducing the Gramian type matrices. Furthermore, we establish some necessary and sufficient conditions of controllability for a class of impulsive switched systems with non-instantaneous jumps on time scales by using the parameter variation method and some Gramian matrices. Moreover, we give some conditions under which the time-invariant impulsive switched system is controllable. Finally, we discuss the controllability results for a class of abstract integro-hybrid evolution systems with impulses on time scales by using the semigroup and evolution operator theory. Several examples have been provided in order to make our theoretical analysis more concrete. In this work, the main techniques used are time scales theory, parameter variation method, Gramian matrices, multivalued map theory, fixed point theorems, semigroup and evolution operator theory.