Existence And Controllability Of Solutions To Some Differential Equations In Banach Spaces

The theory of differential equations in Banach spaces is an important branch of nonlinear analysis, which is applied to many fields, such as partial differential equations, engineering, control theory. For example, by means of semigroup of lin-ear operators, evolution equations can be transformed into differential equations in abstract spaces, which implies that we can deal with partial differential equa-tions in a unified way. Modern control theory is closely related to many branches of mathematics and controllability is the fundamental concept of control theory. Therefore, it has vital theoretical and practical significance to study the existence and controllability of differential equations in Banach spaces.The present dissertation focuses on the impulsive differential equations and fractional differential equations. By using semigroup of linear operators, measure of noncompactness and approximate solutions, we firstly study the existence of solutions to several types of functional differential equations. Then the results are extended to the control problems of differential systems. The present dissertation consists of five chapters.Chapter1introduces briefly the background and our main work.In Chapter2, we discuss the existence of mild solutions to the following nonlocal impulsive differential equations where A is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators{T(t):t≥0}, g is a nonlocal item,Ii are impulsive functions.The tools used in this chapter are semigroup of linear operators, measure of non-compactness and fixed point theorems. In Section2.1, we introduce some concepts and properties of measure of noncompactness. and prove an important property of measure of noncompactness in PC([0, b]; X), which is closely related to impulsive functions (see Lemma2.1.5). In Section2.2, we give the main results of this chap-ter, i.e., the existence of impulsive differential equations are obtained under various conditions. By supposing that semigroup is equicontinuous, we get the existence results of impulsive differential equations when compactness conditions, Lipschitz conditions and mixed-type conditions are satisfied, respectively. In Section2.3, we give an example applied to partial differential equations.Chapter3is concerned with the existence of the impulsive differential inclusions where A is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators{T(t):t≥0}, F is an upper semicontinuous multifunction.We extend the discussions on impulsive differential equations to the scenario of multifunctions. However, the method and focus used in this chapter are different from Chapter2. By supposing the semigroup is compact, we discuss the existence results when nonlocal item g is not compact and not Lipschitz continuous.In Section3.1, we recall some concepts and fixed point theorems on multi-valued analysis. In Section3.2, by constructing the approximate problem of the above impulsive differential inclusions, we get the existence of the impulsive differential inclusions by means of the compactness of approximate solutions set.Chapter4is devoted to the controllability of the following semilinear impulsive differential system where A(t) is a family of linear operators which generates an evolution operator U:Δ={(t, s)∈[0,b] x [0,b]:0<s≤t≤b}â†'L(X), here, X is a Banach space, L(X) is the space of all bounded linear operators in X:B is a bounded linear operator from a Banach space V to X and the control function u(·) is given in L2([0, b], V). By means of measure of noncompactness, we study the exact controllability of impulsive differential system under the noncompact semigroup. By using Monch’s fixed point theorem, we discuss the controllability of the differential system when the evolution system U(t,s) is equicontinuous (see Theorem4.2.1). Furthermore, we construct a new type of noncompact measure and also get the controllability of the above differential system only supposing the evolution system is strongly continuous, without any compactness and equicontinuity hypotheses to evolution system (see Theorem4.2.2). Here, we essentially improve the method in Chapter2. where the semigroup is supposed to be equicontinuous.Chapter5is concerned with the approximate controllability of the following semilinear fractional differential equations where the state variable x(·) takes values in the Hilbert space X:Dq is the Caputo fractional derivative of order q with0<q≤1; A:D(A)(?)Xâ†'X is the infinitesimal generator of a strongly continuous semigroup T(t) on a Hilbert space X;the control function u(·) is given in L2(J. U), U is a Hilbert space:B is a bounded linear operator from U into X.Due to system errors and technical errors in reality, approximate controllability can be applied more widely. By means of semigroup of bounded linear operators and fractional calculus, we discuss the approximate controllability of the above semilin-ear fractional differential equations. We give the definition of the mild solutions to the fractional differential system and get the approximate controllability results with the assumption that the associated linear control system is approximately con-trollable. In our results, the compactness condition and Lipschitz condition to the nonlocal function g are not needed.