The Control Problems Of Systems Governed By Parabolic Partial Differential Equations

Since 1960's, the control problem of distributed parameter systems has been one of the main subjects concerned and studied in the control field. It has strong applied background and becomes increasingly mature and abundant. The control systems governed by parabolic equations are important parts of distributed parameter systems. To study them can not only pose more challenges to mathematical theories, but also deepen comprehension of control theory..The main purpose of this dissertation is to study stabilization of systems governed by parabolic equations and exact controllability of this kind of systems. By studying stabilization of a class of parabolic equations with applied background, we find that for different steady-state solutions of this system one should add different controllers to the system such that these steady-state solutions become locally asymptotic stable. Moreover, as a topic related to the dissertation, we obtain the locally exact controllability of this system.In this dissertation, we mainly use some new methods and techniques of optimal control theory that recently appear. These mainly include the methodto studyasymptotic behavior of system under finite dimension feedback controlby combining state space decomposition with operator Riccati equation, and the method to study exact controllability via Carleman estimate. Meanwhile,we alsoemploy some tools and means from functional analysis and partialdifferential equation, which include semigroups of linear operators, fractional powers of closed operators, interpolation of operator, Sobolev space theory, L~p existence theory of linear parabolic boundary value problems, and method of subsolutions and supersolutions.Compared with the past-related works, the most important feature of the dissertation is to use different controllers at different steady-state solutions when dealing with stabilization. Our results not only further develop and perfect the theory of distributed parameter systems, but also possess obvious applied significance.