Numerical Methods For Ellptic Optimal Control Problem

The optimal control problem can be understood as solving the allowable control variables that can minimize the performance index when the state equation is satisfied and the control task target set can be reached.The application of numerical calculation method in solving the optimal control problem has always been a widely studied topic.The optimal control problem constrained by partial differential equations is applied in engineering,medicine,biology,industry and other fields,such as hydrodynamics,cancer cell tracking and metal smelting process.In this paper,we study the theoretical properties of solving elliptic optimal control problems by using finite difference method,conjugate gradient method and Newton method,and verify our theoretical results by numerical experiments.The specific work of this paper is as follows:For the linear elliptic optimal control problem,we propose a conjugate gradient method with strong Wolfe Powell line search criterion.According to the first-order variation of the functional,we take the definite gradient and construct the conjugate gradient.In the iterative process,the line search condition is used to limit the value of the iterative step,and finally the conjugate gradient method for solving the distributed control problem is constructed.The algorithm has low complexity and simple calculation,and is very suitable for solving large-scale problems.We prove the sufficient condition of conjugate gradient descent and the existence of iterative step size.At the same time,we show that the algorithm has global convergence and linear convergence speed,but its convergence speed will decrease with the increase of the number of iterations.Finally,in the numerical experiment,we use the finite difference method to discretize the differential,transform the partial differential equation into the algebraic difference equation,and obtain the discrete optimal control problem.The global convergence and convergence rate of the algorithm are verified by numerical calculation.For the nonlinear elliptic optimal control problem,we propose a globally convergent semi smooth Newton method.Through the minimum principle,the problem of finding the minimum of functional with constraints is transformed into a simultaneous system composed of state equation,dual state equation and variational inequality.On this basis,the first-order necessary optimality conditions of the problem are derived.After using the finite difference separation discretization,we give the oblique differential of the discrete system,determine the calculation format of the iterative direction,and obtain the iterative formula of the semi smooth Newton method for solving the sparse control problem.Then,the iterative step size is limited by non monotonic line search conditions,and a semi smooth Newton method with global convergence is constructed.We prove the global convergence of the algorithm,and it also retains the local superlinear convergence of the original semi smooth Newton method.The convergence property of the algorithm is verified by numerical experiments.At the same time,compared with the semi smooth Newton method,it is found that the algorithm has faster convergence speed in solving some examples.Finally,we combine this method with multigrid method,and the results show that the combined algorithm has high efficiency.After theoretical proof and experimental verification,we draw the following conclusions.Firstly,the introduction of line search criterion will reduce the convergence speed of the algorithm in most cases,because the line search limits the iterative step size.However,when the line search criterion is introduced into the conjugate gradient method,the accuracy of the experimental results is obviously more accurate than the original conjugate gradient method.Moreover,the introduction of line search criterion makes the improved semi smooth Newton method have the property of global convergence.In the experimental part,we verify that the improved method still has high efficiency.