Approximate Controllability And Optimal Control Problems For Evolution Systems With Delays

Approximate controllability and optimal control problems are the first and the most impor-tant topics in the control theory for infinite dimensional systems and the research on these two topics is of great significance. In this dissertation, we apply the theories of operator semigroups, resolvent operators, fractional power operators, fundamental solutions as well as spectral anal-ysis to study the approximate controllability and the optimal control problems for linear and semilinear evolution systems with time delays on Hilbert spaces. More precisely, we explore the sufficient conditions for the approximate controllability and we prove the existence of op-timal controls and time optimal controls for linear and semilinear control systems respectively. Moreover, for the linear systems, we establish under some conditions the optimality conditions, the integral and the pointwise maximum principles and the existence of bang bang controls.The whole dissertation contains seven chapters.In Chapter 1, we introduce briefly some background and motivation as well as the main results of this dissertation.In Chapter 2, by making use of the theory of resolvent operators and the so-called resol-vent condition, we discuss the approximate controllability for a class of semilinear integro-differential evolution systems with finite delay. Here, we study this problem by assuming the compactness of the analytic semigroup instead of that of the resolvent operator. In addition, we prove the approximate controllability for a concrete linear integro-differential system by self-adjoint technique so that we can provide a clear example to show the application of the obtained results. Thus, our results improve greatly the relative existing ones in this area.In Chapters 3 and 4, we construct the theory of fundamental solution for linear and linear neutral evolution systems with infinite delays, respectively. Then, in Chapters 3,4 and 5, by ap-plying the constructed fundamental solution theory, the Laplace transformation arguments and the resolvent condition, we study the approximate controllability for semilinear evolution sys- tems with infinite delay, semilinear neutral evolution systems with infinite delay and semilinear stochastic evolution systems with infinite delay, respectively. All these systems are perturbed by linear bounded operators. Due to the fundamental solution theory, we weaken partly the restric-tions of uniform boundedness for the nonlinear terms. Hence, the obtained results extend the corresponding conditions in literature. Moreover, the theory of fundamental solution can also be applied to investigate other topics in control theory such as optimal control and stabilization of evolution systems. It may be a strong tool to study these problems.Since the problems of approximate controllability and optimal controls for linear delayed evolution systems play a key role in the study of the corresponding problems for semilinear evolution systems with delay, in Chapter 6 of this dissertation, through analyzing the spectral properties of the fundamental solution for the linear neutral system and for its adjoint system re-spectively, we explore the conditions for the approximate controllability and we discuss various optimal control problems for linear neutral evolution systems with infinite delay. This chapter develops and extends the corresponding results on the linear systems with finite delay so that our obtained results have an important theoretical and practical significance.Finally, in Chapter 7, by using the theory of fundamental solution for linear systems with infinite delay constructed in Chapter 3, and by verifying the compactness of solution operator for a semilinear system, we investigate via limit arguments the existence of optimal controls and time optimal controls for semilinear evolution systems with infinite delay. The convex condition of the cost function used commonly in literature is taken off here. Hence, our results extend the existing works in the papers on this topic.