Research On Finite Volume Method For Several Types Of Nonlinear Optimal Control Problems

Optimal control problems governed by partial differential equations are widely used in electronics,chemical industry,biology and other fields.A suitable numerical method plays a key role for those problems.At present,there are many literatures about using finite element method and mixed finite element method to deal with optimal control problems.Compared with traditional finite element method,the computing format of finite volume method is more economical and effective.Therefore,the finite volume method will have unique advantages for solving some optimal control problems.In this paper,we will investigate a priori error estimates of finite volume method for some nonlinear optimal control problems.The paper consists of three parts.In the first part,we main study a quadratic nonlinear elliptic optimal control problem.Firstly,we use Lagrange multiplier method to transform nonlinear elliptic optimal control problem into a continuous optimal system.The state and dual state equations of the optimal system are discretized by finite volume method,the variational inequalities are obtained by using the method of variational discretization.So,we obtainen the semi-discrete scheme of finite volume element approximation for nonlinear elliptic optimal control problem.Then,the nonlinear term of those probiems is linearized by Taylor's expansion.By using the properties of interpolation operator,projection operator and variational inequalities,we obtain the a priori error estimates.Finally,some numerical experiments are listed to verify the theoretical results.In the second part,we discuss the finite volume method of general nonlinear parabolic optimal control problems.Similarly,a semi-discrete finite volume element scheme for those problems is established by using Lagrange multiplier method,finite volume method,and variational discretization.By applying the Gronwall lemma,interpolation operators,properties of variational inequalities,projection operators,and the Cauchy inequalities,a priori error estimate of the finite volume element approximation for the nonlinear parabolic optimal control problem is given.Finally,a numerical example is given to verify the theoretical results.In the last part,we discuss the finite volume method of quadratic nonlinear hyperbolic optimal control problem.Firstly,we establish the finite volume element discretization scheme for the hyperbolic optimal control problem.Secondly,we use the Taylor expansion,the Gronwall lemma,standard orthogonal projection operator,interpolation operator and the Cauchy inequality to handle the semi-discrete finite volume element scheme.Finally,we derive a priori error estimate of the finite volume element approximation solution for the nonlinear hyperbolic optimal control problem.