Optimal Control For Two Parabolic Equations

Recently people more and more "are concerned of parabolic partial differential equation optimal control problems. On the basis of the optimal control theory about variational inequality and distributed parameter system, the paper considers optimal control problem governed by systerm of semi-linear and convection-diffusion equation with variable coefficients.On the basis of presented results, under given initial and boundary conditions, the paper studies some optimal control problem about semi-linear parabolic system. include giving the definition of weak appropriate solution and weak minimizing sequence. According to preliminary knowledge in chapter two, we prove the existence of optimal solution under given space. Futher, selecting appropriate cost function J(·), we derive a prior estimate of solution using penalty method and other theories. Also give the adjoint equation and weak approximate solution. At the same, the paper deals with the optimal control problem of one dimension convection-diffusion equation with variable coefficients. We treat the convection velocity coefficient as the control term, Using the definition for weak solution of parabolic, we differentiating the objective function with respect to the control h, the existence of optimalty solution is established by minimizing sequence method. Directional and a prior estimate is derived and necessary condition which optimal control satisfied are secured. Thus the optimality system consists of state equation, adjiont equation and an elliptic variational inequality.