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VARIATIONAL DISCRETIZATION OF PARABOLIC CONTROL PROBLEMS IN THE PRESENCE OF POINTWISE STATE CONSTRAINTS

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Journal of Computational Mathematics 2011年第1期29卷 1-15页

作者： Klaus Deckelnick michael hinze Institut fiir Analysis und Numerik Otto-von Guericke-Universitiit Magdeburg Universitiitsplatz 2 39106 Magdeburg Germany Schwerpunkt Optimierung und Approximation Universitiit Hamburg Bundesstraape 55 20146 Hamburg Germany

space dimensions. These bounds follow from uniform estimates for the discretization error of the state under natural regularity requirements.

关键词： Parabolic optimal control problem State constraints Error estimates.

VARIATIONAL DISCRETIZATION OF A CONTROL-CONSTRAINED PARABOLIC BANG-BANG OPTIMAL CONTROL PROBLEM

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Journal of Computational Mathematics 2020年第1期38卷 14-40页

作者： Nikolaus von Daniels michael hinze Schwerpunkt Optimierung und Approximation Universitat HamburgBundesstrase 5520146 HamburgGermany

We consider a control-constrained parabolic optimal control problem without Tikhonov term in the tracking *** the numerical treatment,we use variational discretization of its Tikhonov regularization:For the state and the adjoint equation,we apply Petrov-Galerkin schemes in time and usual conforming finite elements in *** prove a-priori estimates for the error between the discretized regularized problem and the limit *** these estimates are not robust if the regularization parameter tends to zero,we establish robust estimates,which--depending on the problem's regularity——enhance the previous *** the special case of bang-bang solutions,these estimates are further improved.A numerical example confirms our analytical findings.

关键词： Optimal control Heat equation Control constraints Finite elements A-priori error estimates Bang-bang controls

VARIATIONAL DISCRETIZATION FOR OPTIMAL CONTROL GOVERNED BY CONVECTION DOMINATED DIFFUSION EQUATIONS

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Journal of Computational Mathematics 2009年第2期27卷 237-253页

作者： michael hinze Ningning Yan Zhaojie Zhou Department of Mathematics University of Hamburg Germany LSEC Institute of Systems Science Academy of Mathematics and Systems Science Chinese Academy of Science Beijing 100190 China

In this paper, we study variational discretization for the constrained optimal control problem governed by convection dominated diffusion equations, where the state equation is approximated by the edge stabilization Galerkin method. A priori error estimates are derived for the state, the adjoint state and the control. Moreover, residual type a posteriori error estimates in the L^2-norm are obtained. Finally, two numerical experiments are presented to illustrate the theoretical results.

关键词： Constrained optimal control problem Convection dominated diffusion equation Edge stabilization Galerkin method Variational discretization A priori error estimate A posteriori error estimate.

WEAK-DUALITY BASED ADAPTIVE FINITE ELEMENT METHODS FOR PDE-CONSTRAINED OPTIMIZATION WITH POINTWISE GRADIENT STATE-CONSTRAINTS

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Journal of Computational Mathematics 2012年第2期30卷 101-123页

作者： M. Hintermuer michael hinze Ronald H.W. Hoppe Department of Mathematics Humboldt-University of Berlin Germany Department of Mathematics University of Hamburg Germany Department of Mathematics University of Augsburg Germany and Department of MathematicsUniversity of Houston USA

Adaptive finite element methods for optimization problems for second order linear el- liptic partial differential equations subject to pointwise constraints on the l2-norm of the gradient of the state are considered. In a weak duality setting, i.e. without assuming a constraint qualification such as the existence of a Slater point, residual based a posteriori error estimators are derived. To overcome the lack in constraint qualification on the continuous level, the weak Fenchel dual is utilized. Several numerical tests illustrate the performance of the proposed error *** subject classification： 65N30, 90C46, 65N50, 49K20, 49N15, 65K10.

关键词： Adaptive finite element method A posteriori errors Dualization Low regu-larity Pointwise gradient constraints State constraints Weak solutions.