| This thesis is devoted to optimal control problem with leading term containing controls. Three kind of problems are considered, including maximum principle, existence of optimal control and relaxed control problem.The thesis is divided into five chapters.In Chapter1, results for classical optimal control problems, i.e. problems for leading term independent of control are reviewed. Then, the background, present situation on optimal control problem with leading term containing controls are introduced. After that, the results and main difficulties in this thesis are explained.In Chapter2, some preliminaries are introduced briefly, including the related results in well-posedness of PDE, two scale convergence, H-convergence, measure theory and measure valued function space.In Chapter3, a class of optimal control problem governed by semilinear hyperbolic equation is considered, the controls contained in the coefficients of state equation. A hyperbolic version of homogenization result is proved with the help of two scale con-vergence. Using this result, a maximum principle is obtained via homogenization spike variation. At last, the main result is extend to the case with state constrain.Chapter4is devoted to existence theory of elliptic optimal control problem. G-closure of leading term is investigate via H-convergence, and a local representation is given. Using this result, an existence theorem is obtained. In this theorem, some kind of convexity condition is assumed. Such assumption is equivalent to the Cesari condition when leading term does not contain controls, which means the existence theorem is an extension of classic result.In Chapter5, the relaxation of optimal control problem in Chapter4is considered. The corresponding relaxed optimal control problem is received by relax both controls and leading term. A further characterization of G-closure and the local representation for admissible set of relaxed control set are obtained. Then, by homogenization spike variation, maximum principle of relaxed version is established. At last, the isotropic case is considered, and a further result is given. |
