Iterative Proper Orthogonal Decomposition Method For Optimal Control Problem Governed By Elliptic-parabolic Systems

The optimal control problem constrained by partial differential equations has attracted great attention,because it plays an important role in science and engineering.This paper focuses on the optimal control of elliptic-parabolic systems.It can be used to describe optimal control problem of fluid flow through deformable porous media and therefore has important biological and chemical applications.For example,optimize flow pressure and study the influence of relevant biological parameters.In addition,elliptic-parabolic systems can also be used to describe mathematical models of lithium-ion batteries.In this paper,the iterative proper orthogonal decomposition method for optimal control problems of elliptic-parabolic systems is studied.Firstly,for the elliptic-parabolic system,the optimal distributed control problem is studied in this paper.The finite element method is used to solve our problem,and the piecewise linear continuous function is used for space discretization and backward Euler method is used for time discretization.Then,by using the first discretization-then optimization approach,the finite element,approximation equation of the co-state variable of the model and the corresponding optimality condition are derived in detail,so as to form the optimal control system of the fully discrete finite element solution.Then,using the standard optimal control theory,a prior error estimation of the finite element approximation solution is analyzed.Using proper orthogonal decomposition(POD)technology,different POD basis functions of the state and co-state variables are established,respectively,on the basis of taking the numerical solutions of some time nodes as snapshots.Since the initial value of the control variable affects the state and co-state variables,the POD basis functions selected for the first time is generally not the optimal ones.Therefore,this paper proposes iterative method to find the optimal POD basis functions,establish iterative POD scheme and obtain the final POD approximation solution.In order to derive the priori error estimation of the state,co-state and control variables,the error estimation between the approximate solution of iterative POD scheme and the exact solution is derived by introducing intermediate variables of the state and co-state variables.And the convergence of the iterative POD scheme is analyzed.Finally,numerical experiments are given to verify the corresponding theoretical results.Based on the optimal distributed control problem of elliptic-parabolic systems,we further study the optimal inner boundary control problem of elliptic-parabolic systems.The research method is similar to the way mentioned above.The optimal control system of the fully discrete finite element scheme is also derived in detail,and the corresponding priori error estima.tes are analyzed.Similarly,the iterative POD scheme for boundary control is established,and the corresponding error estimation and convergence of the scheme are analyzed.Finally,numerical experiments which verify the theoretical results are presented.