Research On The Interpolation Coefficient Hybrid Finite Element Method For Nonlinear Optimal Control Problems In Control Systems

Optimal control problem has a wide range of applications in many areas of the control system which has been used in integrate and design the speed control system,the most fuel-efficient control system,the minimum energy control system,and the linear regulator.For example: to determine an optimal control method so that the spacecraft has one track to another track with minimal fuel consumption,and to explore the minimum power loss in the circuit transmission system.Therefore,The numerical simulation of these optimal control prolems is one of important areas in scientific and engineering computing.Most of these reaearches aim at the mixed finite element method while there doesn't seem to exist much works on theoretical analysis of interpolation coefficients mixed finite element methods.Interpolation coefficients mixed finite element methods have many advantages.It has much significance to extend the mixed finite element method to interpolation coefficients mixed finite element methods for optimal control problems.In this paper,we will investigate a priori error estimates for some nonlinear optimal control problems by interpolation coefficients mixed finite element methods in the control systems.The paper consists of two parts.In the first part,we study the nonlinear elliptic optimal control problems in the control systems.At first,we transform a minimization problem to a coupled system of state equation,co-state equation and a variational inequality.The state and the co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions.Using the interpolation operator for nonlinear problems which were presented by Larsson and Tomee,we transform this problems to a linear systems.Combining some techniques to deal with elliptic equations,we derive a priori error estimates for nonlinear optimal control problems.We obtain a priori error estimates of nonlinear elliptic optimal control problems in the control systems.Finally,we present some numerical examples which confirms our theoretical results.In the second part,we investigate the nonlinear parabolic control problem in the control systems.Garcia et.al.have some earlier work for the error estimates of mixed finite element methods for parabolic equations.However,there doesn't seem to exist much work on theoretical analysis of mixed finite element approximation for parabolic optional control problems.Firstly,we construct the interpolation coefficient mixed finite element discrete scheme of the nonlinear parabolic optimal control problem.The interpolation coefficients are used to deal with the nonlinear term in the equation.The state equation and the dual state equation are approximated by the lowest order Raviart-Thomas mixed finite element method.The control variable is approximated by the piecewise constant function.Some intermediate variables and corresponding error equations are constructed,and the optimal order error estimates of the approximate solutions of the state variables and the control variables are obtained by combining the properties of the interpolation operators and several other projection operators.Then,we briefly introduce fully discrete interpolation coefficients mixed finite element methods,the time t has been discretized by the difference method and the error equation has been constructed.We obtain a priori error estimates of the fully discrete interpolation coefficient mixed finite element solution.Finally,a numerical example is given to verify the theoretical results.