Convergence of a Finite Element Approximation to a State-Constrained Elliptic Control Problem

Deckelnick, Klaus; Hinze, Michael

doi:10.1137/060652361

Cited by 126 publications

(85 citation statements)

References 9 publications

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“…In [9,10,22] error estimates are given for optimal control problems with finitely many state constraints. In Deckelnick and Hinze [15,16] error estimates of order h 1−ε in 2d and h with a ij , a 0 ∈ L ∞ (Ω), a 0 (x) ≥ 0 for almost all x ∈ Ω. Furthermore, there exists some Λ > 0 such that…”

Section: J(u)mentioning

confidence: 99%

New regularity results and improved error estimates for optimal control problems with state constraints

2014

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Abstract. In this paper we are concerned with a distributed optimal control problem governed by an elliptic partial differential equation. State constraints of box type are considered. We show that the Lagrange multiplier associated with the state constraints, which is known to be a measure, is indeed more regular under quite general assumptions. We discretize the problem by continuous piecewise linear finite elements and we are able to prove that, for the case of a linear equation, the order of convergence for the error in L 2 (Ω) of the control variable is h| log h| in dimensions 2 and 3.Mathematics Subject Classification. 49K20, 49M05, 49M25, 65N30, 65N15.

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Section: J(u)mentioning

confidence: 99%

New regularity results and improved error estimates for optimal control problems with state constraints

2014

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“…We point out, though, that efficient discretization methods for measures have been used in the past, cf. for example [11]. Nevertheless, certain results of numerical analysis, such as mesh independence and convergence of interior point methods are not applicable to problems that admit measures as multipliers.…”

Section: Introductionmentioning

confidence: 99%

On regularization methods for the numerical solution of parabolic control problems with pointwise state constraints

Neitzel¹,

Tröltzsch²

2008

ESAIM: COCV

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Abstract. In this paper we study Lavrentiev-type regularization concepts for linear-quadratic parabolic control problems with pointwise state constraints. In the first part, we apply classical Lavrentiev regularization to a problem with distributed control, whereas in the second part, a Lavrentiev-type regularization method based on the adjoint operator is applied to boundary control problems with state constraints in the whole domain. The analysis for both classes of control problems is investigated and numerical tests are conducted. Moreover the method is compared with other numerical techniques.Mathematics Subject Classification. 49K20, 49N10, 49M05.

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“…[4,6,7,15,18,19,23], to the best of the authors knowledge numerical analysis of parabolic optimal control problems with pointwise bounds in space-time for the state has not yet been considered in the literature. In this work we present the numerical analysis for our result of Theorem 4.1 which we already announced in [11].…”

Section: Introductionmentioning

confidence: 99%

Variational Discretization of Parabolic Control Problems in the Presence of Pointwise State Constraints

Deckelnick

Hinze²

2011

JCM

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We consider a parabolic optimal control problem with pointwise state constraints. The optimization problem is approximated by a discrete control problem based on a discretization of the state equation by linear finite elements in space and a discontinuous Galerkin scheme in time. Error bounds for control and state are obtained both in two and three space dimensions. These bounds follow from uniform estimates for the discretization error of the state under natural regularity requirements.Mathematics subject classification: 49J20, 49K20, 35B37

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Convergence of a Finite Element Approximation to a State-Constrained Elliptic Control Problem

Cited by 126 publications

References 9 publications

New regularity results and improved error estimates for optimal control problems with state constraints

New regularity results and improved error estimates for optimal control problems with state constraints

On regularization methods for the numerical solution of parabolic control problems with pointwise state constraints

Variational Discretization of Parabolic Control Problems in the Presence of Pointwise State Constraints

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