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A NEW FINITE ELEMENT APPROXIMATION OF A STATE-CONSTRAINED optimal control problem

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Journal of Computational Mathematics 2009年第1期27卷 97-114页

作者： Wenbin Liu Wei Gong Ningning Yan KBS Institute of Mathematics and Statistics The University of Kent Canterbury CT2 7NF England LSEC Institute of Systems Science Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China

In this paper, we study numerical methods for an optimal control problem with pointwise state constraints. The traditional approaches often need to deal with the deltasingularity in the dual equation, which causes many difficulties in its theoretical analysis and numerical approximation. In our new approach we reformulate the state-constrained optimal control as a constrained minimization problems only involving the state, whose optimality condition is characterized by a fourth order elliptic variational inequality. Then direct numerical algorithms （nonconforming finite element approximation） are proposed for the inequality, and error estimates of the finite element approximation are derived. Numerical experiments illustrate the effectiveness of the new approach.

关键词： optimal control problem State-constraints Fourth order variational inequalities Nonconforming finite element method.

Analysis of Finite Difference Approximations of an optimal control problem in Economics

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Advances in Applied Mathematics and Mechanics 2019年第6期11卷 1358-1375页

作者： Alexander Lapin Shuhua Zhang Sergey Lapin Na Yan Institute of Computational Mathematics and Information Technologies Kazan Federal UniversityKazan 420008Russia Coordinated Innovation Center for Computable Modeling in Management Science Tianjin University of Finance and EconomicsTianjin 300222China Department of Mathematics and Statistics Washington State UniversityPullmanWA 99163USA Business School Nankai UniversityTianjin 300071China

We consider an optimal control problem which serves as a mathematical model for several problems in economics and *** problem is the minimization of a continuous constrained functional governed by a linear parabolic diffusion-advection equation controlled in a coefficient in advection *** additional constraint is non-negativity of a solution of state *** construct and analyze several mesh schemes approximating the formulated problem using finite difference methods in space and in *** these approximations keep the positivity of the solutions to mesh state problem,either unconditionally or under some additional constraints to mesh *** allows us to remove corresponding constraint from the formulation of the discrete problem to simplify its *** on theoretical estimates and numerical results,we draw conclusions about the quality of the proposed mesh schemes.

关键词： Mean field game optimal control problem parabolic diffusion-advection equation finite difference methods

A SPARSE GRID STOCHASTIC COLLOCATION AND FINITE VOLUME ELEMENT METHOD FOR CONSTRAINED optimal control problem GOVERNED BY RANDOM ELLIPTIC EQUATIONS

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Journal of Computational Mathematics 2018年第2期36卷 310-330页

作者： Liang Ge Tongjun Sun School of Mathematical Sciences University of Jinan Jinan Shandong 250022 China School of Mathematics Shandong University Jinan Shandong 250100 China

In this paper, a hybird approximation scheme for an optimal control problem governed by an elliptic equation with random field in its coefficients is considered. The random coefficients are smooth in the physical space and depend on a large number of random variables in the probability space. The necessary and sufficient optimality conditions for the optimal control problem are obtained. The scheme is established to approximate the optimality system through the discretization by using finite volume element method for the spatial space and a sparse grid stochastic collocation method based on the Smolyak approximation for the probability space, respectively. This scheme naturally leads to the discrete solutions of an uncoupled deterministic problem. The existence and uniqueness of the discrete solutions are proved. A priori error estimates are derived for the state, the co-state and the control variables. Numerical examples are presented to illustrate our theoretical results.

关键词： optimal control problem Random elliptic equations Finite volume element,Sparse grid Smolyak approximation A priori error estimates.

STEP-LIKE CONTRAST STRUCTURE OF SINGULARLY PERTURBED optimal control problem＊

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Journal of Computational Mathematics 2012年第1期30卷 2-13页

作者： Mingkang Ni Limeng Wu Department of Mathematics East China Normal UniversityShanghai 200241China

The existence of step-like contrast structure for a class of singularly perturbed optimal control problem is presented by contrast structure theory. By means of direct scheme of boundary function method, we construct the uniformly valid asymptotic solution for the singularly perturbed optimal control problem. As an application, an example is given to illustrate the main result in this paper.

关键词： Singular perturbation optimal control problem Contrast structure.

Meshfree Finite Volume Element Method for Constrained optimal control problem Governed by Random Convection Diffusion Equations

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Communications in Mathematical Research 2020年第2期36卷 229-246页

作者： Liang Ge Wanfang Shen Wenbin Liu School of Mathematical Sciences University of JinanJinanShandong 250022China School of Mathematic and Quantitative Economics Shandong Key Laboratory of Blockchain FinanceShandong University of Finance and EconomicsJinanShandong 250014China Beijing Normal University Beijing 100875 Beijing Normal University–Hong Kong Baptist University United International College ZhuhaiGuangdong519087China.

In this paper,we investigate a stochastic meshfree finite volume element method for an optimal control problem governed by the convection diffusion equations with random *** are two contributions of this ***,we establish a scheme to approximate the optimality system by using the finite volume element method in the physical space and the meshfree method in the probability space,which is competitive for high-dimensional random ***,the a priori error estimates are derived for the state,the co-state and the control *** numerical tests are carried out to confirm the theoretical results and demonstrate the efficiency of the proposed method.

关键词： optimal control problem stochastic convection diffusion equations meshfree method radial basis functions finite volume element

Virtual Element Discretization of optimal control problem Governed by Brinkman Equations

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Engineering（科研） 2023年第2期15卷 114-133页

作者： Yanwei Li School of Mathematics and Statistics Shandong Normal University Jinan China

In this paper, we discuss virtual element method (VEM) approximation of optimal control problem governed by Brinkman equations with control constraints. Based on the polynomial projections and variational discretization of the control variable, we build up the virtual element discrete scheme of the optimal control problem and derive the discrete first order optimality system. A priori error estimates for the state, adjoint state and control variables in L2 and H1 norm are derived. The theoretical findings are illustrated by the numerical experiments.

关键词： Virtual Element Method optimal control problem Brinkman Equations A Priori Error Estimate

A POSTERIORI ERROR ESTIMATE OF FINITE ELEMENT METHOD FOR THE optimal control WITH THE STATIONARY BENARD problem

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Journal of Computational Mathematics 2013年第1期31卷 68-87页

作者： Yanzhen Chang Danping Yang Department of Mathematics Beijing University of Chemical Technology Beijing 100029 China Department of Mathematics East China Normal University Shanghai 200062 China

In this paper, we consider the adaptive finite element approximation for the distributed optimal control associated with the stationary Benard problem under the pointwise control constraint. The states and co-states are approximated by polynomial functions of lowest- order mixed finite element space or piecewise linear functions and control is approximated by piecewise constant functions. We give the a posteriori error estimates for the control, the states and co-states.

关键词： optimal control problem Stationary Benard problem Nonlinear coupled sys-tem A posteriori error estimate.

Gradient Recovery Based Two-Grid Finite Element Method for Parabolic Integro-Differential optimal control problems

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Journal of Applied Mathematics and Physics 2024年第8期12卷 2849-2865页

作者： Miao Yang School of Mathematics Southwest Jiaotong University Chengdu China

In this paper, the optimal control problem of parabolic integro-differential equations is solved by gradient recovery based two-grid finite element method. Piecewise linear functions are used to approximate state and co-state variables, and piecewise constant function is used to approximate control variables. Generally, the optimal conditions for the problem are solved iteratively until the control variable reaches error tolerance. In order to calculate all the variables individually and parallelly, we introduce a gradient recovery based two-grid method. First, we solve the small scaled optimal control problem on coarse grids. Next, we use the gradient recovery technique to recover the gradients of state and co-state variables. Finally, using the recovered variables, we solve the large scaled optimal control problem for all variables independently. Moreover, we estimate priori error for the proposed scheme, and use an example to validate the theoretical results.

关键词： optimal control problem Gradient Recovery Two-Grid Finite Element Method

SUPERCONVERGENCE ANALYSIS OF FINITE ELEMENT METHODS FOR optimal control problemS OF THE STATIONARY B(?)NARD TYPE

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Journal of Computational Mathematics 2008年第5期26卷 660-676页

作者： Yanzhen Chang Danping Yang School of Mathematics and System Science Shandong University Jinan 250100 China Department of Mathematics East China Normal University Shanghai 200062 China

In this paper, we consider the finite element approximation of the distributed optimal control problems of the stationary Benard type under the pointwise control constraint. The states and the co-states are approximated by polynomial functions of lowest-order mixed finite element space or piecewise linear functions and the control is approximated by piecewise constant functions. We give the superconvergence analysis for the control; it is proved that the approximation has a second-order rate of convergence. We further give the superconvergence analysis for the states and the co-states. Then we derive error estimates in L^∞-norm and optimal error estimates in L^2-norm.

关键词： optimal control problem The stationary Benard problem Nonlinear coupled system Finite element approximation Superconvergence.