Controllability For Some Control Systems Governed By Parabolic Equations

In this dissertation,we discuss the controllability for some control systems governed by parabolic equations.Firstly,we study the controllability of two classes of important quasilinear parabolic equations.The abstract theory for quasilinear parabolic equations has already been quite fruitful.However,in the study of the controllability of semilinear parabolic systems with superlinear nonlinearities, we found that theories are not perfect.In particular,as far as we know,Barbu in obtained null L^r(N) -controls with r(N)∈[2,(2(N+2))/(N-2)]if X≥3;and r(N)∈[2,∞) if N=2;r(N)∈[2,∞]if N=1.But this technique can only be applied to the null controllability of the superlinear heat equation when N＜6 (which does not seem to be a natural restriction on N).Again,L^r(N)-estimates of the controls are needed for control cost in industrial applications,etc.In this article,we continue to consider such a problem and obtain that under certainconditions,it is exactly controllable to the trajectories.And we also extend this result to semilinear parabolic systems with superlinear nonlinearities involving gradient terms.In the following,we study the existence of insensitizing controls for a class of linear parabolic equations in unbounded domain.As we all know,it is equivalent to the null controllability of a cascade system governed by two linear parabolic equations.Under some conditions,these systems are null controllable at any time.Finally,we consider an approximate controllability problem for linear parabolic equations with rapidly oscillating coefficients.We show that,asε→0,the approximate control and the corresponding solution converge respectively to the control and to the corresponding solution of the homogenized problem.