Optimal Control Problems Governed By A Parabolic System With K-approximate Periodic Solutions

This dissertation is concerned with several problems related to the optimal control andidentification of intrinsic parameters of a parabolic system. The system is assumed to havethe so-called k-approximate periodic solutions. A k-approximate periodic solution (to bedefined in Chapter 1) of a parabolic system is a special type of solutions of the equation. Itcan be viewed as a perturbation of a periodic Solution in the low frequency part. Our firstobjective of the thesis is to use the Garlerkin method to get the existence and uniquenessof the k-approximate periodic solutions for a quite general parabolic system. We then con-sider the identification problem through the approximate periodic solutions for the so-calledperturbation term or error term in both a linear and a nonlinear parabolic system. We alsoestablish the Pontryagin maximum principle for the optimal control of a semilinear parabolicequation with the k-approximate periodic solutions.Our thesis has four chapters. In Chapter 1, we give a general overview for topics tobe discussed in details in the later chapters. Chapter 2 is concerned with an identificationproblem for the so-called perturbation term or error term in a linear parabolic partial differ-ential equation, through its approximate periodic solutions. Chapter 3 continues our studyin Chapter 2 on the identification problem for the coefficients for the lower order terms in anonlinear parabolic system. In Chapter 4, we study optimal control problems governed bya semilinear parabolic equation with k-approximate periodic solutions.