Adaptive Methods For Optimal Control Problems Governed By Diferential Equations

During the past several decades, all kinds of diferential equations and the optimalcontrol problems governed by diferential equations gain much attention for their wideapplications. There are lots of works applying spectral methods and finite element meth-ods to solve them.In this thesis, the first work is to solve a complex-valued parabolic equation by anadaptive finite element method. Another work of this thesis is to design the adaptivealgorithms for the optimal control problems governed by ordinary diferential equations.In various spectral methods, the pseudospectral method and the spectral element methodare chosen for the second work.The usefulness of adaptive finite elements is very apparent when the exact solutionhas strong, geometrically localized variations. Many works are devoted to the derivationof a posteriori error estimates and the development of adaptive algorithms for the real-valued parabolic problems. This thesis uses an adaptive algorithm to solve the linearparabolic problem whose coefcient, inhomogeneous term and solution can be complex-valued functions. Some error indicators are given for the adaptive algorithm. By theerror indicators, the adaptive algorithm can capture the strong, geometrically localizedvariations of the exact solution by modifying the space meshes and the tine-step sizes.The reliability of the adaptive algorithm is proved.It is noted that the pseudospectral methods usually use the polynomials with thesame degree to approximate the optimal state and control functions. When the functionswhose properties are obviously diferent from each other need to be approximated, thediferent degrees may be better. For the optimal control problems governed by ordinarydiferential equations, an adaptive pseudospectral method is proposed. In the method,the degrees of the polynomials approximating the optimal state and control functionscan be diferent. The degrees are determined by the a posteriori error estimation of theapproximating polynomials. The number of the discrete collocation points is determinedby the max degree. The feasibility and convergence of the discrete problem are proved.The aim of the adaptive method is to save the time cost without sacrificing accuracy.It is well known that the spectral methods may not be the best choice when thesolution is not smooth. For the optimal control problems whose optimal solutions areweakly discontinuous, an adaptive algorithm is designed. The considered problems arediscretized by pseudospectral method and spectral element method. The time interval isdivided into several subintervals. The piecewise polynomials are used to approximate thesolutions of optimal control problems. According to the numerical solutions, the adaptive algorithm divides the subintervals that may contain the weak discontinuous points intonew subintervals, and increases the degrees of the approximating polynomials in somesubintervals where the exact solution may be smooth. The feasibility and convergenceof the discrete problem are demonstrated.