EXISTENCE AND CONTROLLABILITY RESULTS TO SOME FUNCTIONAL DIFFERENTIAL SYSTEMS

Abstract

The work presented in this thesis deals with the investigations of existence, uniqueness and some controllability results for mild and integral solutions to various types of fractional di erential systems in abstract spaces. To deal with such problems some tools which we have used are the semigroup theory of linear operators, concepts of fractional calculus, functional analysis and some suitable xed point theorems. We may divide our work into three major parts. In the rst part (Chapters 3, 4 and 5), the existence and uniqueness of mild solutions for deterministic and stochastic fractional di erential systems are investigated. In order to obtain the desired results, monotonic iterative technique, condensing theorem and Picard type iterations are employed. It is well-known that the concept of controllability is a valuable property of a control system, and it plays a very important role in several control problems in both nite and in nite dimensional spaces. In controllability of a system, we show the existence of a control function which steers to the mild solution of the system from its initial state to the desired nal state, where the initial and nal states may vary over the entire space. There are several concepts related to controllability, such as exact controllability, optimal controllability, trajectory controllability and approximate controllability. In the thesis, we study exact and approximate controllability v results for some fractional di erential systems. Motivated by the above discussion, in the second part (Chapter 6) of the thesis, the exact controllability results are established for some fractional impulsive delay di erential systems using some basic tools of fractional calculus, measure of noncompactness and M onch xed point theorem. In the third part (Chapter 7), some existence, uniqueness and approximate controllability of integral solutions for fractional di erential systems involving Hilfer fractional derivative with non dense domain are discussed in a Banach space. The chapter-wise organization of the thesis is as follows: Chapter 1 contains a brief introduction to the problems which are discussed in later chapters, and provides a motivational background to study the problems discussed in this thesis. Further, it contains a review of relevant literature. Chapter 2 contains some basic concepts of fractional calculus, functional analysis, semigroup theory and stochastic analysis that will be required in the subsequent chapters. In Chapter 3, we obtain some existence and uniqueness results for mild solutions to Sobolev type fractional impulsive di erential systems with fractional order nonlocal conditions by applying monotone iterative technique coupled with the method of lower and upper solutions. The su cient conditions are obtained by measure of noncompactness and generalized Gronwall inequality. Finally, an application is given to illustrate the obtained results. In Chapter 4, the existence and uniqueness results for mild solutions of a abstract multi-term time-fractional stochastic di erential system are investigated. We use the tools of fractional calculus, generalized semigroup theory and stochastic analysis techniques to obtain the desired results. We come up with a new set vi of su cient conditions using standard Picard's iterations on the coe cients in the equations satisfy some non-Lipschitz conditions. Finally, an application is given to illustrate the obtained results. In Chapter 5, some existence and uniqueness results for mild solutions to the multi-term time-fractional di erential systems with not-instantaneous impulses and nite delay are established. We use the tools of Banach xed point theorem and condensing map along with generalization of the semigroup theory for linear operators and fractional calculus to come up with a new set of su cient conditions for the existence and uniqueness of the mild solutions. An illustration is provided at the end of the chapter to demonstrate the established results In Chapter 6, we obtain some exact controllability results for an abstract fractional impulsive quasilinear integro-di erential system with state-dependent delay. We use the concepts of fractional calculus, measure of noncompactness and abstract phase space to come up with a new set of su cient conditions for the exact controllability by using M onch's xed point theorem. At the end, an example is discussed to demonstrate the application of the obtained abstract results. Chapter 7 is concerned with existence and approximate controllability of integral solutions to the systems determined by abstract fractional di erential equations with nondense domain. We establish the existence and uniqueness results of integral solution by generalized Banach contraction principle. Moreover, our approximate controllability results are based on a sequencing technique in which the compactness of semigroup and uniformly boundedness of nonlinear functions are not required. Finally, an application is given to illustrate the obtained results. The relevant references are appended at the end.