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An interior penalty method for distributed optimal control problems governed by the biharmonic operator

COMPUTERS & MATHEMATICS WITH APPLICATIONS 2014年第12期68卷 2205-2221页

作者： Gudi, Thirupathi Nataraj, Neela Porwal, Kamana Indian Inst Sci Dept Math Bangalore 560012 Karnataka India Indian Inst Technol Dept Math Bombay 400076 Maharashtra India

In this paper, a C-0 interior penalty method has been proposed and analyzed for distributed optimal control problems governed by the biharmonic operator. The state and adjoint variables are discretized using continuous piecewise quadratic finite elements while the control variable is discretized using piecewise constant approximations. A priori and a posteriori error estimates are derived for the state, adjoint and control variables under minimal regularity assumptions. Numerical results justify the theoretical results obtained. The a posteriori error estimators are useful in adaptive finite element approximation and the numerical results indicate that the sharp error estimators work efficiently in guiding the mesh refinement. (C) 2014 Elsevier Ltd. All rights reserved.

关键词： distributed optimal control problems Biharmonic operator C-0 interior penalty method a priori error estimate a posteriori error estimate

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Superconvergence and L^(∞)-Error Estimates of the Lowest Order Mixed Methods for distributed optimal control problems Governed by Semilinear Elliptic Equations

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Numerical Mathematics(Theory,Methods and Applications) 2013年第3期6卷 479-498页

作者： Tianliang Hou School of Mathematical Sciences South China Normal UniversityGuangzhou 510631GuangdongChina

In this paper, we investigate the superconvergence property and the L∞-errorestimates of mixed finite element methods for a semilinear elliptic control problem. Thestate and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant *** derive some superconvergence results for the control variable. Moreover, we derive L^(∞)-error estimates both for the control variable and the state variables. Finally, anumerical example is given to demonstrate the theoretical results.

关键词： Semilinear elliptic equations distributed optimal control problems superconvergence L^(∞)-error estimates mixed finite element methods.

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Convergence of distributed optimal control problems governed by elliptic variational inequalities

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2012年第2期53卷 375-393页

作者： Boukrouche, Mahdi Tarzia, Domingo A. Univ Austral FCE CONICET Dept Matemat RA-1950 Rosario Santa Fe Argentina Univ St Etienne PRES Lyon Univ Math Lab LaMUSE EA 3989 F-42023 St Etienne France

First, let u (g) be the unique solution of an elliptic variational inequality with source term g. We establish, in the general case, the error estimate between and for mu a[0,1]. Secondly, we consider a family of distributed optimal control problems governed by elliptic variational inequalities over the internal energy g for each positive heat transfer coefficient h given on a part of the boundary of the domain. For a given cost functional and using some monotony property between u (3)(mu) and u (4)(mu) given in Mignot (J. Funct. Anal. 22:130-185, 1976), we prove the strong convergence of the optimal controls and states associated to this family of distributed optimal control problems governed by elliptic variational inequalities to a limit Dirichlet distributed optimal control problem, governed also by an elliptic variational inequality, when the parameter h goes to infinity. We obtain this convergence without using the adjoint state problem (or the Mignot's conical differentiability) which is a great advantage with respect to the proof given in Gariboldi and Tarzia (Appl. Math. Optim. 47:213-230, 2003), for optimal control problems governed by elliptic variational equalities.

关键词： Elliptic variational inequalities Convex combinations of the solutions distributed optimal control problems Convergence of the optimal controls Obstacle problem Free boundary problems

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Convergence of distributed optimal control problems governed by elliptic variational inequalities

Convergence of distributed optimal control problems governed...

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Workshop on Non linear Optimization, Variational Inequalities and Equilibrium problems

关键词： Elliptic variational inequalities Convex combinations of the solutions distributed optimal control problems Convergence of the optimal controls Obstacle problem Free boundary problems

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A posteriori error estimates of recovery type for distributed convex optimal control problems

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JOURNAL OF SCIENTIFIC COMPUTING 2007年第2期33卷 155-182页

作者： Li, Ruo Liu, Wenbin Yan, Ningning Peking Univ Sch Math Sci Beijing 100871 Peoples R China Univ Kent KBS Inst Math & Stat Canterbury CT2 7NF Kent England Chinese Acad Sci Acad Math & Syst Sci Inst Syst Sci Beijing Peoples R China

In this paper, we derive a posteriori error estimates of recovery type, and present the superconvergence analysis for the finite element approximation of distributed convex optimal control problems. We provide a posteriori error estimates of recovery type for both the control and the state approximation, which are generally equivalent. Under some stronger assumptions, they are further shown to be asymptotically exact. Such estimates, which are apparently not available in the literature, can be used to construct adaptive finite element approximation schemes and as a reliability bound for the control problems. Numerical results demonstrating our theoretical results are also presented in this paper.

关键词： distributed optimal control problems finite element approximation superconvergence a posteriori error estimator

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optimal-control COMPUTATION FOR PARABOLIC-SYSTEMS WITH BOUNDARY-CONDITIONS INVOLVING TIME DELAYS

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1987年第3期53卷 475-507页

作者： WONG, KH 1.Department of Applied Mathematics University of the Witnwatersrand Johannesburg South Africa

In this paper, we consider a class of optimal control problems involving a second-order, linear parabolic partial differential equation with Neumann boundary conditions. The time-delayed arguments are assumed to appear in the boundary conditions. A necessary and sufficient condition for optimality is derived, and an iterative method for solving this optimal control problem is proposed. The convergence property of this iterative method is also investigated.

关键词： Parabolic differential equations second boundary-value problems time-delayed arguments distributed optimal control problems necessary and sufficient conditions existence theory algorithms convergence of algorithms Galerkin's scheme lumped-parameter systems

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optimal control of linear periodic resonant systems in hilbert spaces

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SIAM JOURNAL ON control AND OPTIMIZATION 1997年第6期35卷 2137-2156页

作者： Barbu, V UNIV IASI ROMANIAN ACADIASI 6600ROMANIA

This work concerns existence, the maximum principle, and synthesis for optimal control problems with linear periodic dynamics in Hilbert spaces in the resonant case. Applications to distributed and boundary periodic c... 详细信息

关键词： distributed optimal control problems maximum principle closed range p-stabilizable

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An a posteriori error analysis of adaptive finite element methods for distributed elliptic control problems with control constraints

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ESAIM-control OPTIMISATION AND CALCULUS OF VARIATIONS 2008年第3期14卷 540-560页

作者： Hintermueller, Michael Hoppe, Ronald H. W. Iliash, Yuri Kieweg, Michael Graz Univ Inst Math A-8010 Graz Austria Univ Houston Dept Math Houston TX 77204 USA Univ Augsburg Inst Math D-86159 Augsburg Germany

We present an a posteriori error analysis of adaptive finite element approximations of distributed control problems for second order elliptic boundary value problems under bound constraints on the control. The error analysis is based on a residual-type a posteriori error estimator that consists of edge and element residuals. Since we do not assume any regularity of the data of the problem, the error analysis further invokes data oscillations. We prove reliability and efficiency of the error estimator and provide a bulk criterion for mesh refinement that also takes into account data oscillations and is realized by a greedy algorithm. A detailed documentation of numerical results for selected test problems illustrates the convergence of the adaptive finite element method.

关键词： a posteriori error analysis distributed optimal control problems control constraints adaptive finite element methods residual-type a posteriori error estimators data oscillations

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Adaptive finite element methods for mixed control-state constrained optimal control problems for elliptic boundary value problems

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2010年第3期46卷 511-533页

作者： Hoppe, R. H. W. Kieweg, M. Univ Augsburg Inst Math D-86159 Augsburg Germany Univ Houston Dept Math Houston TX 77204 USA

Mixed control-state constraints are used as a relaxation of originally state constrained optimal control problems for partial differential equations to avoid the intrinsic difficulties arising from measure-valued multipliers in the case of pure state constraints. In particular, numerical solution techniques known from the pure control constrained case such as active set strategies and interior-point methods can be used in an appropriately modified way. However, the residual-type a posteriori error estimators developed for the pure control constrained case can not be applied directly. It is the essence of this paper to show that instead one has to resort to that type of estimators known from the pure state constrained case. Up to data oscillations and consistency error terms, they provide efficient and reliable estimates for the discretization errors in the state, a regularized adjoint state, and the control. A documentation of numerical results is given to illustrate the performance of the estimators.

关键词： distributed optimal control problems Mixed control-state constraints Adaptive finite elements Posteriori error analysis

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A residual based snapshot location strategy for POD in distributed optimal control of linear parabolic equations

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IFAC-PapersOnLine 2016年第8期49卷 13-18页

作者： Alla, A. Gräßle, C. Hinze, M. Florida State University Department of Scientific Computing 400 Dirac Science Library Tallahassee FL32306 United States Universität Hamburg Bundesstr. 55 Hamburg20146 Germany

In this paper we study the approximation of a distributed optimal control problem for linear parabolic PDEs with model order reduction based on Proper Orthogonal Decomposition (POD-MOR). POD-MOR is a Galerkin approach where the basis functions are obtained upon information contained in time snapshots of the parabolic PDE related to given input data. In the present work we show that for POD-MOR in optimal control of parabolic equations it is important to have knowledge about the controlled system at the right time instances. For the determination of the time instances (snapshot locations) we propose an a-posteriori error control concept which is based on a reformulation of the optimality system of the underlying optimal control problem as a second order in time and fourth order in space elliptic system. Finally, we present numerical tests to illustrate our approach and to show the effectiveness of the method in comparison to existing approaches. © 2016

关键词： Location Numerical methods optimal control systems Partial differential equations Principal component analysis distributed optimal control distributed optimal control problems Linear parabolic equation Model order reduction optimal control problem optimal controls Posteriori error control Proper orthogonal decompositions

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