Comparison of optimality systems for the optimal control of the obstacle problem

Harder, Felix; Wachsmuth, Gerd

doi:10.1002/gamm.201740004

Cited by 30 publications

(25 citation statements)

References 32 publications

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“…A similar eect cannot be observed in the time-independent case, cf., e.g., the results in [22,33]. Note that (5.13) is still a necessary optimality condition when H 1 0 -q.e.…”

Section: Directional Dierentiability Of the Solution Map Having Estamentioning

confidence: 89%

Sensitivity Analysis and Optimal Control of Obstacle-Type Evolution Variational Inequalities

Christof¹

2019

SIAM J. Control Optim.

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This paper is concerned with the dierential sensitivity analysis and the optimal control of evolution variational inequalities (EVIs) of obstacle type. We demonstrate by means of a counterexample that the solution map S of an EVI with a unilateral constraint is typically not (weakly) directionally dierentiable or Lipschitz continuous in any of the spaces H s (0, T ; H), s ≥ 1/2, where (0, T) is the time interval and H is the pivot space of the underlying Gelfand triple V → H → V *. We further establish that, despite this negative result, the solution operator is always strongly Hadamard directionally dierentiable as a function S : L 2 (0, T ; H) → L q (0, T ; H) for all 1 ≤ q < ∞, weakly-directionally dierentiable as a function S : L 2 (0, T ; H) → L ∞ (0, T ; H), and weakly directionally dierentiable as a function S : L 2 (0, T ; H) → L 2 (0, T ; V). Using the dierentiability properties of the map S, we derive strong stationarity conditions for optimal control problems that are governed by EVIs of obstacle type. The resulting optimality system is compared with that obtained by regularization.

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“…A similar eect cannot be observed in the time-independent case, cf., e.g., the results in [22,33]. Note that (5.13) is still a necessary optimality condition when H 1 0 -q.e.…”

Section: Directional Dierentiability Of the Solution Map Having Estamentioning

confidence: 89%

Sensitivity Analysis and Optimal Control of Obstacle-Type Evolution Variational Inequalities

Christof¹

2019

SIAM J. Control Optim.

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“…in Ω, then local minimizersū ∈ L 2 (0, T ; L 2 (Ω)) of (P) are also solutions of a stationarity system (roughly of weak type, cf. [34] and [56], Sect. 4.1) which involves an adjoint statep ∈ L ∞ (0, T ; L 2 (Ω)) ∩ L 2 (0, T ; H 1 0 (Ω)) ∩ BV ([0, T ]; Y * γ ) and a multiplierμ ∈ M([0, T ] × cl(Ω)) ∩ W 0 (0, T ) * .…”

Section: Introductionmentioning

confidence: 99%

New regularity results and finite element error estimates for a class of parabolic optimal control problems with pointwise state constraints

Christof

Vexler

2021

ESAIM: COCV

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We study first-order necessary optimality conditions and finite element error estimates for a class of distributed parabolic optimal control problems with pointwise state constraints. It is demonstrated that, if the bound in the state constraint and the differential operator in the governing PDE fulfill a certain compatibility assumption, then locally optimal controls satisfy a stationarity system that allows to significantly improve known regularity results for adjoint states and Lagrange multipliers in the parabolic setting. In contrast to classical approaches to first-order necessary optimality conditions for state-constrained problems, the main arguments of our analysis require neither a Slater point, nor uniform control constraints, nor differentiability of the objective function, nor a restriction of the spatial dimension. As an application of the established improved regularity properties, we derive new finite element error estimates for the dG(0)-cG(1)-discretization of a purely state-constrained linear-quadratic optimal control problem governed by the heat equation. The paper concludes with numerical experiments that confirm our theoretical findings.

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“…A limit analysis for vanishing regularization then yields an optimality system for the original non-smooth problem which is usually of intermediate strength and less rigorous compared to strong stationarity. A classification of the different optimality systems involving dual variables for the case of optimal control of the obstacle problem can be found in [14,10]. In [26,15,40], optimality conditions for the case of VIs of the first kind are obtained by using limiting normal cones.…”

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confidence: 99%

“…In [26,15,40], optimality conditions for the case of VIs of the first kind are obtained by using limiting normal cones. As shown in [10,Theorem 5.7], this approach does in general not lead to optimality conditions which are more rigorous than what is obtained by regularization (even if the limiting normal cone in the spirit of Mordukhovich is used). On the other hand, it is known for the case of finite-dimensional mathematical programs with equilibrium constraints that under suitable assumptions, the optimality conditions obtained via regularization are equivalent to zero being in the Clarke subdifferential of the reduced objective; see, e.g., [4,Section 2.3.3].…”

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confidence: 99%

Optimal control of a non-smooth semilinear elliptic equation

Christof¹,

Meyer²,

Walther³

et al. 2018

Mathematical Control &Amp; Related Fields

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This paper is concerned with an optimal control problem governed by a non-smooth semilinear elliptic equation. We show that the control-to-state mapping is directionally differentiable and precisely characterize its Bouligand subdifferential. By means of a suitable regularization, first-order optimality conditions including an adjoint equation are derived and afterwards interpreted in light of the previously obtained characterization. In addition, the directional derivative of the control-to-state mapping is used to establish strong stationarity conditions. While the latter conditions are shown to be stronger, we demonstrate by numerical examples that the former conditions are amenable to numerical solution using a semi-smooth Newton method.

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Comparison of optimality systems for the optimal control of the obstacle problem

Abstract: We consider stationarity systems for the optimal control of the obstacle problem. The focus is on the comparison of several different systems which are provided in the literature. We obtain some novel results concerning the relations between these stationarity concepts.

Cited by 30 publications

References 32 publications

Sensitivity Analysis and Optimal Control of Obstacle-Type Evolution Variational Inequalities

Sensitivity Analysis and Optimal Control of Obstacle-Type Evolution Variational Inequalities

New regularity results and finite element error estimates for a class of parabolic optimal control problems with pointwise state constraints

Optimal control of a non-smooth semilinear elliptic equation

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