APPROXIMATE CONTROLLABILITY OF SEMILINEAR IMPULSIVE FUNCTIONAL EVOLUTION EQUATIONS

Abstract

Controllability of dynamical systems governed by partial di↵erential equations has been a widely active research area in the past few decades. Controllability is a qualitative property of dynamical systems and plays a significant role in mathematical control theory. A systematic study of controllability was started in the early sixties of the last century. Generally, a system’s controllability (exact controllability) refers to steering a system from any initial state to an arbitrary desired final state in finite time constrained to an admissible class of controls. In the study of infinite dimensional controlled dynamical systems, the concept of exact controllability and approximate controllability are prominent. In the case of infinite dimensional control systems, the problem of exact controllability rarely holds. The assumptions required to produce the exact controllability results are too restrictive and are extremely difficult to verify in practical situations. Consequently, it is necessary to examine the weak notion of controllability known as approximate controllability. Approximate controllability means a system can lead from any initial state to an arbitrarily small neighborhood of a desired final state. The problem of approximate controllability is extensively studied, as it is relatively sustainable and plausible from the application’s perspective. The main objective of the thesis is to study the existence and approximate controllability results for infinite dimensional semilinear deterministic impulsive control systems or inclusions with delays (finite or infinite) in a separable reflexive Banach space having uniform convex dual . To achieve our aims, we have used the fixed point approach and resolvent operator technique along with semigroup theory, evolution family, cosine and sine families, and fractional calculus. The results can be applied in a class of impulsive functional evolution systems appearing in the mathematical models of several physical phenomena. Construction of the thesis is as follows: Chapter 1 contains the introductory materials along with literature survey on the controllability of linear, semilinear and nonlinear control systems of integer and fractional orders. Chapter 2 briefly provides the mathematical tools required to understand the main chapters. Certain topics of functional analysis and fractional calculus are discussed. Moreover, the semigroup theory, evolution family, cosine and sine families, and controllability results for linear and semilinear systems are given, which will be useful for the subsequent chapters. Chapter 3 is devoted to the approximate controllability of a time-variant impulsive system consisting of state-dependent delay in a separable Banach space, which is reflexive. Firstly, we discuss the approximate controllability of the linear problem corresponding to the semilinear equation. Then we formulate sufficient conditions for the approximate controllability of the impulsive semilinear systems by utilizing the evolution family, the fractional power operator theory, properties of the resolvent operator, and Schauder’s fixed point theorem. We verify the existence of a mild solution by invoking Darbo’s fixed point theorem. In the end, we provide two concrete examples to validate our results. Approximate controllability of certain Sobolev type semilinear impulsive evolution equations with finite delay is established in Chapter 4. The approximate controllability of the linear control system corresponding to the semilinear Sobolev type impulsive system is discussed in details. For this, we first formulate a linear regulator problem to obtain the existence of the optimal control. Then we derive the explicit expression of the optimal control in the feedback form. Using this feedback control, we achieve the approximate controllability of the linear control system. Further, we establish our main result, that is, the approximate controllability of the Sobolev type impulsive system with finite delay. To complete this aim, we first prove the existence of a mild solution for the system by invoking Schauder’s fixed point theorem. Then, we obtain the approximate controllability of that system by employing the resolvent operator condition. Finally, we discuss a concrete example to illustrate the efficiency of the developed results. Existence of a mild solution and approximate controllability results for a class of impulsive functional control systems involving time-dependent operators in Banach spaces are developed in Chapter 5. We employ a generalization of the Leray-Schauder fixed point theorem for multi-valued maps along with the evolution family to obtain the existence of a mild solution. Then we derive sufficient conditions for the approximate controllability of the considered problem using the resolvent operator condition. Finally, an example is provided to demonstrate the application of the constructed theory. Chapter 6 concentrates on the approximate controllability of the control problems governed by fractional non-instantaneous impulsive functional evolution equations with state-dependent delay involving Caputo fractional derivatives in Banach spaces. We study the approximate controllability of a semilinear functional impulsive system by invoking the resolvent operator technique and Schauder’s fixed point theorem. A proper motivation for constructing di↵erent forms of feedback controls for the fractional order semilinear systems available in the literature is justified. Moreover, we rectify some shortcomings of the existing works in the context of characterization of phase space and approximate controllability of the fractional order impulsive systems in Banach spaces. Eventually, we apply these results to obtain the approximate controllability of a fractional order heat equation with non-instantaneous impulses and delay. Chapter 7 deals with the approximate controllability problem for fractional impulsive functional evolution equations of order ↵ 2 (1, 2) in Banach spaces. A new set of conditions for the existence and approximate controllability of the considered system have been developed by applying the concept of ↵/2− resolvent family (related to the strongly continuous cosine family generated by the operator A) along with the resolvent operator condition and Schauder’s fixed point theorem. The construction of a feedback control and approximate controllability of the linear fractional control system of order 1 < ↵ < 2 are discussed in details. We also establish the existence of a mild solution to the semilinear system via Krasnoselskii’s fixed point theorem. Moreover, we also provide a remark on further relaxations of the conditions on nonlinear function f(·, ·) as well as impulse functions. In the end, a concrete example is discussed to explain the developed theory. In Chapter 8, we study a stronger version of the approximate controllability known as the finite-approximate controllability. Finite-approximate refers to a final state of the control system that satisfies the approximate controllability condition and simultaneously exact controllability condition in finite dimensional spaces under some control function This chapter contains three sections. The first section describes a useful characterization of finite-approximate controllability for linear fractional evolution equations of order 1 < ↵ < 2 in terms of the resolvent-like operator. We construct a suitable control to find the approximate controllability of the linear control system. This control also ensures the finite-approximate controllability of the same linear problem. Further, we describe the finite-approximate controllability results of a semilinear impulsive fractional evolution equation. The second section concerns the finite-approximate controllability of fractional evolution equations of order 1 < ↵ < 2 with finite delay. Sufficient conditions for the finite-approximate controllability are formulated by extending the variational method. In the final section, we apply the demonstrated results to a fractional wave equation involving impulses and delay. At the end of the thesis, some conclusion about the works are presented. A short discussion of possible future directions is also given.