Second-Order and Stability Analysis for State-Constrained Elliptic Optimal Control Problems with Sparse Controls

Casas, Eduardo; Tröltzsch, Fredi

doi:10.1137/130917314

Cited by 24 publications

(24 citation statements)

References 18 publications

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“…Several publications followed, investigating semi-linear elliptic equations, parabolic linear, and parabolic semi-linear equations; we refer for instance to Refs. [35][36][37] among others.…”

Section: Sparse Optimal Controlmentioning

confidence: 99%

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Analytical, Optimal, and Sparse Optimal Control of Traveling Wave Solutions to Reaction-Diffusion Systems

Ryll

Löber

Martens

et al. 2016

Understanding Complex Systems

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Abstract. This work deals with the position control of selected patterns in reactiondiffusion systems. Exemplarily, the Schlögl and FitzHugh-Nagumo model are discussed using three different approaches. First, an analytical solution is proposed. Second, the standard optimal control procedure is applied. The third approach extends standard optimal control to so-called sparse optimal control that results in very localized control signals and allows the analysis of second order optimality conditions.

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Section: Sparse Optimal Controlmentioning

confidence: 99%

“…Therefore, it is also called sparse control or sparse optimal control in the literature, see Refs. [34][35][36][37]. In some sense, we can interpret the areas with non-vanishing sparse optimal control signals as the most sensitive areas of the RD patterns with respect to the desired control goal.…”

Section: Introductionmentioning

confidence: 99%

Analytical, Optimal, and Sparse Optimal Control of Traveling Wave Solutions to Reaction-Diffusion Systems

Ryll

Löber

Martens

et al. 2016

Understanding Complex Systems

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“…where C 1 is as in (5), the solution of problem EP is independent of and solves the optimal control problem EC with the terminal constraint x.T / D 0, defined as EC:…”

Section: Epmentioning

confidence: 99%

Exact penalization of terminal constraints for optimal control problems

Gugat

Zuazua

2016

Optim. Control Appl. Meth.

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Summary We study optimal control problems for linear systems with prescribed initial and terminal states. We analyze the exact penalization of the terminal constraints. We show that for systems that are exactly controllable, the norm‐minimal exact control can be computed as the solution of an optimization problem without terminal constraint but with a nonsmooth penalization of the end conditions in the objective function, if the penalty parameter is sufficiently large. We describe the application of the method for hyperbolic and parabolic systems of partial differential equations, considering the wave and heat equations as particular examples. Copyright © 2016 John Wiley & Sons, Ltd.

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“…To enforce sparsity of the controls, i.e., the localization of the controls in a small region of the domain, we modify our functional in a way to include the L 1 (Ω T ) norm. It is well understood that the inclusion of the L 1 norm in the cost functional yields sparse controls (see, for instance, [6,7,13,20,23]). In [6] necessary and sufficient second order optimality conditions are derived for a semilinear elliptic control problem.…”

Section: Eduardo Casas and Konstantinos Chrysafinosmentioning

confidence: 99%

Analysis of the Velocity Tracking Control Problem for the 3D Evolutionary Navier--Stokes Equations

Casas¹,

Chrysafinos²

2016

SIAM J. Control Optim.

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Abstract. The velocity tracking problem for the evolutionary Navier-Stokes equations in three dimensions is studied. The controls are of distributed type and are submitted to bound constraints. The classical cost functional is modified so that a full analysis of the control problem is possible. First and second order necessary and sufficient optimality conditions are proved. A fully discrete scheme based on a discontinuous (in time) Galerkin approach, combined with conforming finite element subspaces in space, is proposed and analyzed. Provided that the time and space discretization parameters, τ and h, respectively, satisfy τ ≤ Ch 2 , the L 2 (Ω T ) error estimates of order O(h) are proved for the difference between the locally optimal controls and their discrete approximations. Finally, combining these techniques and the approach of Casas, Herzog, and Wachsmuth [SIAM J. Optim., 22 (2012), pp. 795-820], we extend our results to the case of L 1 (Ω T ) type functionals that allow sparse controls.

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Second-Order and Stability Analysis for State-Constrained Elliptic Optimal Control Problems with Sparse Controls

Cited by 24 publications

References 18 publications

Analytical, Optimal, and Sparse Optimal Control of Traveling Wave Solutions to Reaction-Diffusion Systems

Analytical, Optimal, and Sparse Optimal Control of Traveling Wave Solutions to Reaction-Diffusion Systems

Exact penalization of terminal constraints for optimal control problems

Analysis of the Velocity Tracking Control Problem for the 3D Evolutionary Navier--Stokes Equations

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