On Some Iterative Methods For Optimal Control Problems

The research on the constrained optimal control of partial differential equations is of great significance to physics,engineering and other fields.In this paper,we consider the constrained optimal control problem of elliptic partial differential equations and study the iterative method for large-scale saddle point systems obtained by discretization of elliptic partial differential equations.Aiming at the classical Uzawa algorithm,we obtain the explicit expression of the optimal relaxation parameter of the iterative algorithm using Fourier analysis,and analyze numerically the effectiveness of the parameter.Furthermore,according to the characteristics of the discrete saddle point problem,we propose the state control alternating iterative Method(SCAIM).On the algebraic level,we prove the convergence of SCAIM algorithm and give the optimal selection of relaxation parameter in the SCAIM algorithm by means of Fourier analysis.At the same time,we give the convergence rate estimation of SCAIM algorithm,whose performance does not depend on the mesh size.In addition,the optimal relaxation parameter selection of SCAIM algorithm has nothing to do with the discretization method since we use the continuous level analysis.Numerical experiments show that when SCAIM is used as a precondition,its algorithm performance is superior to that of the precondition based on Schur complement.