scholarly journals Galerkin spectral method for elliptic optimal control problem with $L^2$-norm control constraint

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Zhen-Zhen Tao ◽  
Bing Sun
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Error Estimates ◽  
A Posteriori Error Estimates ◽  
Control Variable ◽  
Approximation Problem ◽  
Posteriori Error ◽  
Spectral Approximation ◽  
Norm Error

<p style='text-indent:20px;'>This paper is concerned with the Galerkin spectral approximation of an optimal control problem governed by the elliptic partial differential equations (PDEs). Its objective functional depends on the control variable governed by the <inline-formula><tex-math id="M1">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm constraint. The optimality conditions for both the optimal control problem and its corresponding spectral approximation problem are given, successively. Thanks to some lemmas and the auxiliary systems, a priori error estimates of the Galerkin spectral approximation problem are established in detail. Moreover, a posteriori error estimates of the spectral approximation problem are also investigated, which include not only <inline-formula><tex-math id="M2">\begin{document}$ H^1 $\end{document}</tex-math></inline-formula>-norm error for the state and co-state but also <inline-formula><tex-math id="M3">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm error for the control, state and costate. Finally, three numerical examples are executed to demonstrate the errors decay exponentially fast.</p>

scholarly journals A posteriori error estimates of hp spectral element method for parabolic optimal control problems

2022 ◽  
Vol 7 (4) ◽  
pp. 5220-5240
Author(s):  
Zuliang Lu ◽  
◽  
Fei Cai ◽  
Ruixiang Xu ◽  
Chunjuan Hou ◽  
...  
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Error Estimates ◽  
A Posteriori Error Estimates ◽  
Control Variable ◽  
Posteriori Error ◽  
Spectral Element ◽  
Element Approximation ◽  
Parabolic Optimal Control

<abstract><p>In this paper, we investigate the spectral element approximation for the optimal control problem of parabolic equation, and present a hp spectral element approximation scheme for the parabolic optimal control problem. For improve the accuracy of the algorithm and construct an adaptive finite element approximation. Under the Scott-Zhang type quasi-interpolation operator, a $ L^2(H^1)-L^2(L^2) $ posteriori error estimates of the hp spectral element approximated solutions for both the state variables and the control variable are obtained. Adopting two auxiliary equations and stability results, a $ L^2(L^2)-L^2(L^2) $ posteriori error estimates are derived for the hp spectral element approximation of optimal parabolic control problem.</p></abstract>

scholarly journals Adaptive Finite Element Method for Optimal Control Problem Governed by Linear Quasiparabolic Integrodifferential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-26 ◽  
Author(s):  
Wanfang Shen ◽  
Hua Su
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
A Posteriori Error Estimates ◽  
Mathematical Formulation ◽  
Posteriori Error ◽  
Adaptive Finite Element Method ◽  
Adaptive Finite Element ◽  
A Posteriori Error ◽  
Quadratic Optimal Control

The mathematical formulation for a quadratic optimal control problem governed by a linear quasiparabolic integrodifferential equation is studied. The control constrains are given in an integral sense:Uad={u∈X;∫ΩUu⩾0,t∈[0,T]}. Then the a posteriori error estimates inL∞(0,T;H1(Ω))-norm andL2(0,T;L2(Ω))-norm for both the state and the control approximation are given.

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.

Morley FEM for a Distributed Optimal Control Problem Governed by the von Kármán Equations

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Sudipto Chowdhury ◽  
Neela Nataraj ◽  
Devika Shylaja
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Error Estimates ◽  
Lower Order ◽  
Control Variable ◽  
The State ◽  
Distributed Optimal Control ◽  
Von Kármán Equations ◽  
Adjoint Variables

AbstractConsider the distributed optimal control problem governed by the von Kármán equations defined on a polygonal domain of {\mathbb{R}^{2}} that describe the deflection of very thin plates with box constraints on the control variable. This article discusses a numerical approximation of the problem that employs the Morley nonconforming finite element method (FEM) to discretize the state and adjoint variables. The control is discretized using piecewise constants. A priori error estimates are derived for the state, adjoint and control variables under minimal regularity assumptions on the exact solution. Error estimates in lower-order norms for the state and adjoint variables are derived. The lower-order estimates for the adjoint variable and a post-processing of control leads to an improved error estimate for the control variable. Numerical results confirm the theoretical results obtained.

Dynamical Behavior of a Malaria Model with Discrete Delay and Optimal Insecticide Control

2017 ◽  
Vol 12 (01) ◽  
pp. 19-38 ◽  
Author(s):  
Tuhin Kumar Kar ◽  
Soovoojeet Jana
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Basic Reproduction Number ◽  
Human Population ◽  
Reproduction Number ◽  
Dynamical Behavior ◽  
Control Variable ◽  
Discrete Delay ◽  
Insecticide Control

In this paper we have proposed and analyzed a simple three-dimensional mathematical model related to malaria disease. We consider three state variables associated with susceptible human population, infected human population and infected mosquitoes, respectively. A discrete delay parameter has been incorporated to take account of the time of incubation period with infected mosquitoes. We consider the effect of insecticide control, which is applied to the mosquitoes. Basic reproduction number is figured out for the proposed model and it is shown that when this threshold is less than unity then the system moves to the disease-free state whereas for higher values other than unity, the system would tend to an endemic state. On the other hand if we consider the system with delay, then there may exist some cases where the endemic equilibrium would be unstable although the numerical value of basic reproduction number may be greater than one. We formulate and solve the optimal control problem by considering insecticide as the control variable. Optimal control problem assures to obtain better result than the noncontrol situation. Numerical illustrations are provided in support of the theoretical results.

scholarly journals Ana posteriorierror analysis for an optimal control problem with point sources

2018 ◽  
Vol 52 (5) ◽  
pp. 1617-1650 ◽  
Author(s):  
Alejandro Allendes ◽  
Enrique Otárola ◽  
Richard Rankin ◽  
Abner J. Salgado
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Weighted Sobolev Spaces ◽  
Control Variable ◽  
The State ◽  
Point Sources ◽  
Adjoint Equations ◽  
Error Estimator ◽  
A Posteriori

We propose and analyze a reliable and efficienta posteriorierror estimator for a control-constrained linear-quadratic optimal control problem involving Dirac measures; the control variable corresponds to the amplitude of forces modeled as point sources. The proposeda posteriorierror estimator is defined as the sum of two contributions, which are associated with the state and adjoint equations. The estimator associated with the state equation is based on Muckenhoupt weighted Sobolev spaces, while the one associated with the adjoint is in the maximum norm and allows for unbounded right hand sides. The analysis is valid for two and three-dimensional domains. On the basis of the deviseda posteriorierror estimator, we design a simple adaptive strategy that yields optimal rates of convergence for the numerical examples that we perform.

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