Second Order Analysis for Strong Solutions in the Optimal Control of Parabolic Equations

Bayen, Térence; Silva, Francisco

doi:10.1137/141000415

Cited by 22 publications

(24 citation statements)

References 19 publications

(37 reference statements)

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“…Although our results are similar to the ones of [1], [2], they are more general than those of [1], [2]. In particular, we admit the case of a vanishing Tikhonov regularization parameter ν, while ν > 0 is required in [1], [2].…”

Section: Introductionsupporting

confidence: 77%

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Second-Order Optimality Conditions for Weak and Strong Local Solutions of Parabolic Optimal Control Problems

Casas

Tröltzsch

2015

Vietnam J. Math.

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Second order sufficient optimality conditions are considered for a simplified class of semilinear parabolic equations with quadratic objective functional including distributed and terminal observation. Main emphasis is laid on problems where the objective functional does not include a Tikhonov regularization term. Here, standard second order conditions cannot be expected to hold. For this case, new second order conditions are established that are based on different types of critical cone. Depending on the choice of this cone, the second order conditions are sufficient for local minima that are weak or strong in the sense of calculus of variations.

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Section: Introductionsupporting

confidence: 77%

“…Moreover, an unknown referee called our attention to the preprint [2] on a parabolic control problem.…”

Section: Introductionmentioning

confidence: 99%

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Second-Order Optimality Conditions for Weak and Strong Local Solutions of Parabolic Optimal Control Problems

Casas

Tröltzsch

2015

Vietnam J. Math.

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“…It is described in the textbooks [14,27] and was developed in the last two decades. We invite the reader to refer to [5,7,22,26,3] and the references therein on this topic. The main difficulty in the derivation of sufficient optimality conditions, called two-norm discrepancy, lies in the fact that the cost function is twice differentiable for the L ∞ -norm, but one can only assume that its Hessian is coercive for the L 2 -norm on an appropriate subspace.…”

Section: Introductionmentioning

confidence: 99%

Hybrid optimal control problems for a class of semilinear parabolic equations

Court¹,

Kunisch²,

Pfeiffer³

2018

Discrete &Amp; Continuous Dynamical Systems - S

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A class of optimal control problems of hybrid nature governed by semilinear parabolic equations is considered. These problems involve the optimization of switching times at which the dynamics, the integral cost, and the bounds on the control may change. First-and secondorder optimality conditions are derived. The analysis is based on a reformulation involving a judiciously chosen transformation of the time domains. For autonomous systems and timeindependent integral cost, we prove that the Hamiltonian is constant in time when evaluated along the optimal controls and trajectories. A numerical example is provided.

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“…where Cū is the cone Cū = {v ∈ L 2 (Q) satisfying the sign condition (2) and J ′ (ū; v) = 0}, v(x, t) ≥ 0 ifū(x, t) = α, ≤ 0 ifū(x, t) = β.…”

mentioning

confidence: 99%

Critical Cones for Sufficient Second Order Conditions in PDE Constrained Optimization

Casas¹,

Mateos²

2020

SIAM J. Optim.

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In this paper, we analyze optimal control problems governed by semilinear parabolic equations. Box constraints for the controls are imposed and the cost functional involves the state and possibly a sparsity-promoting term, but not a Tikhonov regularization term. Unlike finite dimensional optimization or control problems involving Tikhonov regularization, second order sufficient optimality conditions for the control problems we deal with must be imposed in a cone larger than the one used to obtain necessary conditions. Different extensions of this cone have been proposed in the literature for different kinds of minima: strong or weak minimizers for optimal control problems. After a discussion on these extensions, we propose a new extended cone smaller than those considered until now. We prove that a second order condition based on this new cone is sufficient for a strong local minimum.

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Second Order Analysis for Strong Solutions in the Optimal Control of Parabolic Equations

Cited by 22 publications

References 19 publications

Second-Order Optimality Conditions for Weak and Strong Local Solutions of Parabolic Optimal Control Problems

Second-Order Optimality Conditions for Weak and Strong Local Solutions of Parabolic Optimal Control Problems

Hybrid optimal control problems for a class of semilinear parabolic equations

Critical Cones for Sufficient Second Order Conditions in PDE Constrained Optimization

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