scholarly journals Error estimates in $ L^2 $ and $ L^\infty $ norms of finite volume method for the bilinear elliptic optimal control problem

2021 ◽  
Vol 6 (8) ◽  
pp. 8585-8599
Author(s):  
Zuliang Lu ◽  
◽  
Xiankui Wu ◽  
Fei Cai ◽  
Fei Huang ◽  
...  
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Finite Volume Method ◽  
Error Estimates ◽  
Finite Volume ◽  
Elliptic Optimal Control Problem ◽  
Elliptic Optimal Control ◽  
Volume Method

Numerical algorithms for solving an elliptic optimal control problem with a power-law nonlinearity

2021 ◽  
Vol 26 (jai2021.26(2)) ◽  
pp. 64-76
Author(s):  
Hart L ◽  
◽  
Yatsechko N ◽  
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Piecewise Linear ◽  
Boundary Value ◽  
Method Of Lines ◽  
The Method Of Lines ◽  
Gradient Type ◽  
Elliptic Optimal Control Problem ◽  
Elliptic Optimal Control

The paper is devoted to the development and analysis of approximation-iteration algorithms based on the method of grids and the method of lines for solving an elliptic optimal control problem with a power-law nonlinearity. For the numerical solution of the main boundary value problem and the adjoint one, the second order of accuracy difference schemes are applied using the implicit method of simple iteration. Computational schemes of the method of lines for solving the above-mentioned elliptic boundary value problems are implemented in combination with the shooting method for the approximate solution of boundary value problems for the corresponding ordinary differential equations systems arising in the considered domain after lattice approximation. To minimize the objective functional, well-known gradient-type methods (gradient projection and conditional gradient methods) of constrained optimization are used. The essence of the proposed approximation-iteration approach consists in replacing the original extremal problem with a sequence of grid problems that approximate it on a set of refining grids, and applying an iterative gradient-type method to each of the "approximate" extremal problems. In this case, we propose to construct only a few approximations to the solution for each of the "approximate" problems and to take the last of these approximations, using piecewise linear interpolation, as the initial approximation in the iterative process for the next "approximate" problem. The sequence of the corresponding piecewise linear interpolants is considered as a sequence of approximations to the solution of the original extremal problem. The paper discusses the theoretical foundations of this combined approach, as well as its advantages over traditional methods using the example of solving a model optimal control problem

scholarly journals P 1 finite element methods for an elliptic optimal control problem with pointwise state constraints

2018 ◽  
Vol 40 (1) ◽  
pp. 1-28 ◽  
Author(s):  
Susanne C Brenner ◽  
Li-yeng Sung ◽  
Joscha Gedicke
Keyword(s):  
Optimal Control ◽  
Finite Element ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Finite Element Methods ◽  
State Constraints ◽  
Distributed Optimal Control ◽  
Pointwise State Constraints ◽  
Elliptic Optimal Control Problem ◽  
Elliptic Optimal Control

Abstract We present theoretical and numerical results for two $P_1$ finite element methods for an elliptic distributed optimal control problem on general polygonal/polyhedral domains with pointwise state constraints.

scholarly journals Error estimates of finite volume method for Stokes optimal control problem

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Lin Lan ◽  
Ri-hui Chen ◽  
Xiao-dong Wang ◽  
Chen-xia Ma ◽  
Hao-nan Fu
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Error Estimates ◽  
Finite Volume ◽  
Stokes Equations ◽  
A Priori ◽  
The State ◽  
Element Approximation ◽  
Norm Error

AbstractIn this paper, we discuss a priori error estimates for the finite volume element approximation of optimal control problem governed by Stokes equations. Under some reasonable assumptions, we obtain optimal $L^{2}$ L 2 -norm error estimates. The approximate orders for the state, costate, and control variables are $O(h^{2})$ O ( h 2 ) in the sense of $L^{2}$ L 2 -norm. Furthermore, we derive $H^{1}$ H 1 -norm error estimates for the state and costate variables. Finally, we give some conclusions and future works.

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