Numerical approximation of the LQR problem in a strongly damped wave equation

Hernández, Erwin; Kalise, Dante; Otárola, Enrique

doi:10.1007/s10589-008-9213-6

Cited by 4 publications

(4 citation statements)

References 14 publications

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“…In the second test we focus on the undamped case β = 0. Both examples are compared with a linear-quadratic regulator (LQR) controller implemented as in [10]. This latter approach is based on the solution of a large-scale Riccati equation and does not consider a dimensional reduction step.…”

Section: Numerical Testsmentioning

confidence: 99%

“…Although a detailed picture of these contributions goes beyond the goals of this paper we want to mention some recent and relevant contributions on controllability by Lasiecka and Triggiani [16], Privat, Trélat and Zuazua [17] and Zhang, Zheng and Zuazua [20]. A thorough analysis of the approximation of the linear-quadratic regulator problem for the wave equation can be found in [10]. The interested reader can also find a comprehensive presentation of control problems for waves and their approximation in [6] (see also the long list of references therein).…”

Section: Introduction and Description Of The Problemmentioning

confidence: 99%

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A HJB-POD feedback synthesis approach for the wave equation

Alla

Falcone

Kalise

2016

Bull Braz Math Soc, New Series

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We propose a computational approach for the solution of an optimal control problem governed by the wave equation. We aim at obtaining approximate feedback laws by means of the application of the dynamic programming principle. Since this methodology is only applicable for low-dimensional dynamical systems, we first introduce a reduced-order model for the wave equation by means of Proper Orthogonal Decomposition. The coupling between the reduced-order model and the related dynamic programming equation allows to obtain the desired approximation of the feedback law. We discuss numerical aspects of the feedback synthesis and providenumerical tests illustrating this approach

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Section: Numerical Testsmentioning

confidence: 99%

Section: Introduction and Description Of The Problemmentioning

confidence: 99%

A HJB-POD feedback synthesis approach for the wave equation

Alla

Falcone

Kalise

2016

Bull Braz Math Soc, New Series

Self Cite

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“…The optimization problem (1.8)-(1.9) is of relevance in applications where one can specify a control at a finitely many pre-specified points. For instance, references [7,48] discuss applications within the context of the active control of sound and [26,34,35] in the active control of vibrations; see also [24,29,44]. An analysis of problem (1.8)-(1.10) is presented in [30], where the authors use the variational discretization concept to derive error estimates.…”

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confidence: 99%

Some applications of weighted norm inequalities to the error analysis of PDE-constrained optimization problems

Antil

Otárola

Salgado

2017

IMA Journal of Numerical Analysis

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The purpose of this work is to illustrate how the theory of Muckenhoupt weights, Muckenhoupt weighted Sobolev spaces and the corresponding weighted norm inequalities can be used in the analysis and discretization of PDE constrained optimization problems. We consider: a linear quadratic constrained optimization problem where the state solves a nonuniformly elliptic equation; a problem where the cost involves pointwise observations of the state and one where the state has singular sources, e.g. point masses. For all three examples we propose and analyze numerical schemes and provide error estimates in two and three dimensions. While some of these problems might have been considered before in the literature, our approach allows for a simpler, Hilbert space-based, analysis and discretization and further generalizations.

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“…For existing results in the context of feedback control of the wave equation we refer to [19], for the strongly damped wave equation and the Timoshenko beam to [25,24]. For open-loop control of the wave equation see [18,23,29,30,32] and the references cited therein.…”

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confidence: 99%

Optimal feedback control for undamped wave equations by solving a HJB equation

2015

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Abstract. An optimal finite-time horizon feedback control problem for (semi-linear) wave equations is presented. The feedback law can be derived from the dynamic programming principle and requires to solve the evolutionary Hamilton-Jacobi-Bellman (HJB) equation. Classical discretization methods based on finite elements lead to approximated problems governed by ODEs in high dimensional spaces which makes the numerical resolution by the HJB approach infeasible. In the present paper, an approximation based on spectral elements is used to discretize the wave equation. The effect of noise is considered and numerical simulations are presented to show the relevance of the approach.

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Numerical approximation of the LQR problem in a strongly damped wave equation

Abstract: Optimal control, Feedback control, Wave equation, Convergence rates, Finite element method,

Cited by 4 publications

References 14 publications

A HJB-POD feedback synthesis approach for the wave equation

A HJB-POD feedback synthesis approach for the wave equation

Some applications of weighted norm inequalities to the error analysis of PDE-constrained optimization problems

Optimal feedback control for undamped wave equations by solving a HJB equation

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