Finite Element Method For Optimal Control Problems With Integral Constraint And Parameter Estimation

Optimal control of partial differential equations is a very active interdisciplinary subject in mathematical science.Most of the optimal control problems can be expressed by the following abstract mathematical models:min{J(u,y)}u?Uod such that A(y,u)=0.Where u is control variable,y is state variable,Uad called a constrained control set,and A(y,u)is partial differential equation which we call it a state equation.The application of finite element method in the optimal control of partial differential equations has been extensively and deeply studied,and it has abundant achievements in convergence and error analysis or numerical calculation.Among many types of finite element methods,adaptive finite element method(AFEM)has become a main method in engineering calculation and discipline because it can greatly improve the computational efficiency.The implementation mechanism of the adaptive finite element method is that the meshes are refined in the region with large error estimates,so that the meshes are densely distributed where the regularity of the function is poor.For this reason,the estimator of posterior error is very important for the reliability and effectiveness of the adaptive method.In Chapter 2,we studied the elliptic optimal control problem with integral constraints.By using the a priori error argument,the optimal convergence order of priori errors of finite element solutions for this type of problems in the sense of L2 and L? norm is obtained,and our theoretical deduction conclusion is proved by numerical simulation experiments.In Chapter 3,we studied the parabolic optimal control problem with integral constraints.The state variables of the problem are approximated by linear continuous functions,while the control variable is approximated by piecewise constants,finally,the a priori error estimates of control and state variable approximation are given.In Chapter 4,the adaptive finite element method for a class of optimal control problems with parameter estimation is studied,the sharp a posteriori error estimate is developed,and the error estimators are deduced mainly by L?(0,T;L2(?))and L2(0,T;H1(?))norm.