A Priori Error Analysis for Discretization of Sparse Elliptic Optimal Control Problems in Measure Space

Pieper, Konstantin; Vexler, Boris

doi:10.1137/120889137

Cited by 50 publications

(68 citation statements)

References 17 publications

(37 reference statements)

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“…where the last equivalence is due to a result by Pieper and Vexler [27]. Finally, the inequalities 1…”

Section: Lemma 32 Let μ ∈ M(ω) Be a Positive Measure With A Compactmentioning

confidence: 79%

“…For the case A = −Δ this lemma is proven in [27]. For the general case, let us consider the solution…”

Section: Lemma 32 Let μ ∈ M(ω) Be a Positive Measure With A Compactmentioning

confidence: 93%

“…However, no additional regularity for the adjoint stateφ and the Lagrange multiplier μ was obtained there. Our regularity result has been inspired by a recent result by Pieper and Vexler [27] for a sparse control problem with controls in a measure space.…”

Section: J(u)mentioning

confidence: 96%

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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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“…where the last equivalence is due to a result by Pieper and Vexler [27]. Finally, the inequalities 1…”

Section: Lemma 32 Let μ ∈ M(ω) Be a Positive Measure With A Compactmentioning

confidence: 79%

“…For the case A = −Δ this lemma is proven in [27]. For the general case, let us consider the solution…”

Section: Lemma 32 Let μ ∈ M(ω) Be a Positive Measure With A Compactmentioning

confidence: 93%

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

2014

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“…The approach was extended to semi-linear elliptic equations in [3]. A priori error estimates for finite element approximation of linear elliptic optimal control problems with measure valued controls were investigated in [16]. The parabolic case was considered in [6,13] with different measure valued topologies to enhance directional spatial sparsity.…”

Section: Introductionmentioning

confidence: 99%

Parabolic control problems in space-time measure spaces

Casas

Kunisch

2016

ESAIM: COCV

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Abstract. Optimal control problems in measure spaces governed by parabolic equations with are considered. The controls appear as spatial measure in the initial condition and as space-time measures as forcing functions. First order optimality conditions are derived and certain structural properties, in particular sparsity, are discussed. An framework for approximation if these highly irregular problems is also proposed.Mathematics Subject Classification. 90C48, 49J52, 49K20.

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“…Also in optimal control with partial differential equation constraints, it became rather popular to use L 1 -minimization to enforce sparsity of controls [18][19][20][21][22][23][24], for instance in the modelling of optimal placing of actuators or sensors. In order to give precise meaning to the limit of the optimal control problems (1.9)-(1.10) for the number N of followers tending to infinity, we need to address a few technical challenges.…”

Section: Introductionmentioning

confidence: 99%

Mean-field sparse optimal control

Fornasier

Piccoli

Rossi

2014

Phil. Trans. R. Soc. A.

102

122

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We introduce the rigorous limit process connecting finite dimensional sparse optimal control problems with ODE constraints, modelling parsimonious interventions on the dynamics of a moving population divided into leaders and followers, to an infinite dimensional optimal control problem with a constraint given by a system of ODE for the leaders coupled with a PDE of Vlasov-type, governing the dynamics of the probability distribution of the followers. In the classical mean-field theory, one studies the behaviour of a large number of small individuals freely interacting with each other, by simplifying the effect of all the other individuals on any given individual by a single averaged effect. In this paper, we address instead the situation where the leaders are actually influenced also by an external policy maker , and we propagate its effect for the number N of followers going to infinity. The technical derivation of the sparse mean-field optimal control is realized by the simultaneous development of the mean-field limit of the equations governing the followers dynamics together with the Γ -limit of the finite dimensional sparse optimal control problems.

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A Priori Error Analysis for Discretization of Sparse Elliptic Optimal Control Problems in Measure Space

Cited by 50 publications

References 17 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

Parabolic control problems in space-time measure spaces

Mean-field sparse optimal control

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