Approximate Controllability Of Several Kinds Of Second Order Functional Differential Systems

The problem of approximate controllability of second order functional evolution systems is an important issue of control theory for infinite-dimensional evolution equations.The research of this topic is very meaningful and has great applications.In this dissertation,based on the basic theory of partial functional differential equations,cosine operators family and theory of stochastic analysis,we study the existence and uniqueness of mild solutions for several kinds of second order evolution equations with delay and their approximate controllability problems.The whole thesis contains five chapters.In Chapter 1,we introduce some research backgrounds and the significance of the research on evolution equations with delay and the controllability problems.We also present some recent relevant works on functional evolution systems and their controllability.In the end we state briefly the main work of this dissertation.In Chapter 2,we first establish the theory of fundamental solutions for the associated linear second order evolution systems with finite delay,from which the expressions of mild solutions of semilinear second order functional evolution systems are obtained by the technique of Laplace transform.Then using Schauder fixed point theorem we prove the existence and uniqueness of mild solutions for the semilinear systems.Based on this we show the approximate controllability of the considered systems through the so-called resolvent conditions and the compactness of the family of sine operators.The theory of functional differential equations with state-dependent delay is one of the focused area of the research upon functional differential equations in recent years.Based on the foundation of theory of fundamental solutions for linear second order evolution systems with infinite delay,in Chapter 3 of this thesis we discuss the existence and uniqueness of mild solutions for semilinear second-order evolution equations with state-dependent delay in Hilbert space and prove the approximate controllability of the considered systems.Particularly,for the case of the nonlinear terms in the systems involving partial derivatives of spatial variables,we apply the theory of fractional power operators to study the existence and uniqueness of mild solutions of semilinear systems by using fixed point theorem in fractional power subspace and achieve some sufficient conditions of approximate controllability for the system.In Chapter 4 and Chapter 5 of this dissertation,by making use of the theory of fundamental solutions on linear second order equations with infinite delay established in Chapter 3 and combining theory of cosine operators and theory of phase spaces,we exploit the problems of approximate controllability respectively for two kinds of semilinear second order stochastic evolution systems with infinite delay driven by Wiener process and L(?)vy process.We first certify the existence and uniqueness of mild solutions for the stochastic evolution systems by using the Banach contraction principle and the related theory of stochastic analysis.We then study the approximate controllability of the stochastic evolution systems by utilizing the resolvent type conditions,the compactness of sine operators and the uniform boundedness of the nonlinear terms and obtain successfully some sufficient conditions of approximate controllability.Examples to illustrate the applications of the obtained results are provided as well in the end.