Numerical Method For Optimal Control Problem Of A Class Of Maxwell Equation

Electromagnetics plays an important role in many modern applications and technologies.They include applications in energy science,nanotechnology,life science,magnetic confinement fusion,magnetic levitation technology,microwave heating,sensor technology,and many more.Some practical problems in physics or engineering can be reduced to optimal control problems constrained by Maxwell’s equations.However,the structure of the optimal control problem of the Maxwell equation in the time domain is very complex,and it is quite difficult to directly obtain its explicit solution by analytic method.Therefore,it is necessary to study its numerical solution.This paper discusses a numerical method of a class of Maxwell equations,and applies this method to numerically solve the corresponding optimal control problem of the Maxwell equation,as follows:Firstly,the finite difference time domain(FDTD)scheme and the fourth-order difference scheme of this kind of Maxwell equation are given,and the convergence of these two methods is analyzed.A numerical example is given.The results of the example are consistent with the theory,that is,the FDTD scheme is second-order convergence in time and space,the fourth-order difference scheme is second-order convergence in time and fourth-order convergence in space.Secondly,the FDTD scheme and the fourth-order difference scheme are applied to the optimal control problem of the time-domain Maxwell equation.Through the Lagrange multiplier technique and the Helmholtz decomposition,the optimal system composed of state equation,dual state equation,and first-order optimality condition is obtained.On this basis,two calculation formats are given,the iterative method of calculation is established,and a numerical example is given.Numerical results show that the proposed method is effective and feasible.