Receding horizon control for the stabilization of the wave equation

Azmi, Behzad; Kunisch, Karl

doi:10.3934/dcds.2018021

Cited by 11 publications

(14 citation statements)

References 45 publications

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“…It follows that ŷ (1) (τ ) Y ≤ M 0 e λτ b n ≤ M 0 e λτ0 b n = rb n . As a direct consequence of (63), we have ŷ (2)…”

Section: Error Estimate For the Rhc Algorithmmentioning

confidence: 82%

“…To this end, we proceed as in the proof of Lemma 15. We consider the solutions (ŷ (1) ,û (1) ,p (1) ) and (ŷ (2) ,û (2) ,p (2) ) to the linear system (OS), with parameters (y n −ȳ n , T,Π(T − t n ), 0) and (0, T,Π(T − t n ), w), respectively, so that (ŷ,û,p) = (ŷ (1) ,û (1) ,p (1) ) + (ŷ (2) ,û (2) ,p (2) ). Let us first apply Theorem 2 to the first system (with µ = 0).…”

Section: Error Estimate For the Rhc Algorithmmentioning

confidence: 99%

“…Let us fix ε = 1 (2M ) 2 , where M is the constant obtained in the last inequality. It follows that there exists (another) constant M > 0 such that…”

Section: Case Without Weightmentioning

confidence: 99%

“…Let us point at some references from the large literature on receding horizon control. For finite-dimensional systems, we mention [10,16], for infinite-dimensional systems, we mention [1,2,8], and for discrete-time systems the articles [7,11].…”

Section: Introduction 1contextmentioning

confidence: 99%

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On the Turnpike Property and the Receding-Horizon Method for Linear-Quadratic Optimal Control Problems

Breiten¹,

Pfeiffer²

2020

SIAM J. Control Optim.

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Optimal control problems with a very large time horizon can be tackled with the Receding Horizon Control (RHC) method, which consists in solving a sequence of optimal control problems with small prediction horizon. The main result of this article is the proof of the exponential convergence (with respect to the prediction horizon) of the control generated by the RHC method towards the exact solution of the problem. The result is established for a class of infinite-dimensional linear-quadratic optimal control problems with time-independent dynamics and integral cost. Such problems satisfy the turnpike property: the optimal trajectory remains most of the time very close to the solution to the associated static optimization problem. Specific terminal cost functions, derived from the Lagrange multiplier associated with the static optimization problem, are employed in the implementation of the RHC method.

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“…It follows that ŷ (1) (τ ) Y ≤ M 0 e λτ b n ≤ M 0 e λτ0 b n = rb n . As a direct consequence of (63), we have ŷ (2)…”

Section: Error Estimate For the Rhc Algorithmmentioning

confidence: 82%

Section: Error Estimate For the Rhc Algorithmmentioning

confidence: 99%

“…Let us fix ε = 1 (2M ) 2 , where M is the constant obtained in the last inequality. It follows that there exists (another) constant M > 0 such that…”

Section: Case Without Weightmentioning

confidence: 99%

Section: Introduction 1contextmentioning

confidence: 99%

See 2 more Smart Citations

On the Turnpike Property and the Receding-Horizon Method for Linear-Quadratic Optimal Control Problems

Breiten¹,

Pfeiffer²

2020

SIAM J. Control Optim.

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show abstract

“…This may not be realistic from a practical point of view. A first way to circumvent, in the spirit of [4] is to split the time interval into a finite number of sub-intervals and then to compute optimal control on each subinterval.…”

Section: Command Lawmentioning

confidence: 99%

Modeling and optimizing a road de-icing device by a nonlinear heating

Bernardin

Münch

2019

ESAIM: M2AN

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In order to design a road de-icing device by heating, we consider in the one dimensional setting the optimal control of a parabolic equation with a nonlinear boundary condition of the Stefan–Boltzmann type. Both the punctual control and the corresponding state are subjected to a unilateral constraint. This control problem models the heating of a road during a winter period to keep the road surface temperature above a given threshold. The one-dimensional modeling used in this work is a first step of the modeling of a road heating device through the circulation of a coolant in a porous layer of the road. We first prove, under realistic physical assumptions, the well-posedness of the direct problem and the optimal control problem. We then perform some numerical experiments using real data obtained from experimental measurements. This model and the corresponding numerical results allow to quantify the minimal energy to be provided to keep the road surface without frost or snow.

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A framework for randomized time-splitting in linear-quadratic optimal control

Veldman

Zuazua

2022

Numer. Math.

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Inspired by the successes of stochastic algorithms in the training of deep neural networks and the simulation of interacting particle systems, we propose and analyze a framework for randomized time-splitting in linear-quadratic optimal control. In our proposed framework, the linear dynamics of the original problem is replaced by a randomized dynamics. To obtain the randomized dynamics, the system matrix is split into simpler submatrices and the time interval of interest is split into subintervals. The randomized dynamics is then found by selecting randomly one or more submatrices in each subinterval. We show that the dynamics, the minimal values of the cost functional, and the optimal control obtained with the proposed randomized time-splitting method converge in expectation to their analogues in the original problem when the time grid is refined. The derived convergence rates are validated in several numerical experiments. Our numerical results also indicate that the proposed method can lead to a reduction in computational cost for the simulation and optimal control of large-scale linear dynamical systems.

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Receding horizon control for the stabilization of the wave equation

Cited by 11 publications

References 45 publications

On the Turnpike Property and the Receding-Horizon Method for Linear-Quadratic Optimal Control Problems

On the Turnpike Property and the Receding-Horizon Method for Linear-Quadratic Optimal Control Problems

Modeling and optimizing a road de-icing device by a nonlinear heating

A framework for randomized time-splitting in linear-quadratic optimal control

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