Some Studies On Variational Discretization For Two Kinds Of Optimal Control Problems

Optimal control problems have been widely used in many kinds of practicalproblems, such as, atmospheric pollution control, temperature control, oil exploit-ing, image processing, etc. Most of the optimal control problems need to be solvedquickly and the computational works are very large [83], so it is very importantto study efcient numerical methods for these problems. So far, there are follow-ing numerical methods can be used to solve optimal control problems: standardfnite element method, mixed fnite element method, least-squares method, spec-tral method, multigrid method, SPQ method, etc. It is well known that fniteelement method is one of the most efcient and widely used and investigated nu-merical methods. However, for a constrained optimal control problem [55,56], theregularity of the control variable is low. Usually, the control is approximated bypiecewise constant functions and the state and the adjoint state are approximatedby continuous piecewise linear functions in standard fnite element method. Thena priori error estimates of the control is the frst order. The one point fve orderor the second order superconvergence results can be obtained by post-processingmethod. But this will add the computational cost. Recently, Hinze propose avariational discrete conception. By using this conception to solve optimal controlproblems with control constraints, we can not only improve a priori error estimatesof the control to the second order but also save the computational work. Hence,this method is very signifcant.In this paper, we mainly study variational discretization for the following twokinds of optimal control problems:In the frst part, we study a nonlinear elliptic optimal control problems. Ac-cording to the variational principle and optimization theory, we get the equivalentoptimality conditions of the original problems. Due to the regularity of the controlis lower than the regularity of the state and the adjoint state, we use variationaldiscretization to solve this problem, namely we discretize the space of the statevariable and the adjoint state variable, and not directly discretize the space ofthe control variable, by introducing a pointwise projection operator and using theimplicit relationship between the control and the adjoint state, we can get the nu-merical solution of the control, the state and the adjoint state. From the convexity of the objective functional, the error estimates of the fnite element interpolationand the Aubin-Nitsche technique, we derive a priori error estimates of variationaldiscretization for the nonlinear elliptic optimal control problems. Further more, weobtain a posteriori error estimates of residual type by using the Bubble function.We also do some numerical experiments to verify our theoretical results.In the second part, we discuss a linear parabolic optimal control problems.We use isometric division and backward Euler method for the approximation ofthe time variable and use quasi-uniform triangle mesh and linear standard fniteelement method for the approximation of the space variable. First, we derivea priori error estimates of variational discretization for parabolic optimal controlproblems. Secondly, we obtain the superconvergence properties between the ellipticprojection of the state and the adjoint state and the numerical solutions. Finally,we establish a posteriori error estimates of residual type of variational discretizationfor the parabolic optimal control problem. A lot of numerical examples indicatethat our numerical results are consistent of our theoretical results.