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

Casas, Eduardo; Chrysafinos, Konstantinos

doi:10.1137/140978107

Cited by 26 publications

(21 citation statements)

References 17 publications

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“…For several issues related to the analysis and numerics of optimal control problems we refer the reader to [49] (see also references within). Finally, we refer the reader to [9] for the analysis of control problems of 3D evolution NavierStokes equations.…”

Section: Related Resultsmentioning

confidence: 99%

A Review of Numerical Analysis for the Discretization of the Velocity Tracking Problem

Casas

Chrysafinos

2016

SEMA SIMAI Springer Series

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In this paper we are reviewing results regarding the velocity tracking problem. In particular, we focus on our work (Casas and Chrysafinos, SIAM J. Numer. Anal. 50(5): 2281-2306, 2012 Casas and Chrysafinos, Numer. Math. 130:615-643, 2015; and Casas and Crysafinos, to appear in ESAIM: COCV) concerning a-priori error estimates for the velocity tracking of two-dimensional evolutionary Navier-Stokes flows. The controls are of distributed type, and subject to point-wise control constraints. The standard tracking type functional is considered, however the option of setting the penalty-regularization parameter D 0 in front of the L 2 .0; TI L 2 .˝// norm of the control in the functional is also discussed. The discretization scheme of the state and adjoint equations is based on a discontinuous time-stepping scheme combined with conforming finite elements (in space) for the velocity and pressure. Provided that the time and space discretization parameters, and h respectively, satisfy Ä p / for some p > 2, are discussed for the difference between the locally optimal controls and their discrete approximations, when the controls are discretized by piecewise constants functions, the variational discretization approach or by using piecewise-linears in space respectively for > 0. For the case of D 0, (bangbang type controls) we also discuss various issues related to the analysis and discretization, emphasizing on the different features compared to the case > 0. In particular, fully-discrete estimates for the states are presented and discussed.

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Section: Related Resultsmentioning

confidence: 99%

A Review of Numerical Analysis for the Discretization of the Velocity Tracking Problem

Casas

Chrysafinos

2016

SEMA SIMAI Springer Series

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“…Following [6], Theorem 4.2.1, since v ∈ L 2 (L 2 ) and n d ∈ X → L 10/3 (L 10/3 ), there exists a unique solution u ∈ X 2 → L 10 (L 10 ) of (4.14) 3 . This impliesū • ∇c ∈ L 20/3 (L 20/3 ) and we can follow the ideas in the proof of Proposition 3.5 to prove the existence of a unique solution [n,c] ∈ X × X.…”

Section: Regularity Criterionmentioning

confidence: 99%

An optimal control problem related to a 3D-chemotaxis-Navier-Stokes model

López-Ríos

Villamizar-Roa

2021

ESAIM: COCV

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In this paper, we study an optimal control problem associated to a 3D-chemotaxis-Navier-Stokes model. First we prove the existence of global weak solutions of the state equations with a linear reaction term on the chemical concentration equation, and an external source on the velocity equation, both acting as controls on the system. Second, we establish aregularity criterion to get global-in-time strong solutions. Finally, we prove the existence of an optimal solution, and we establish a first-order optimality condition.

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“…The reader is referred to [11] for the proofs of the results of this section. Related references are [5] and [14]. These papers are devoted to the control of the Navier-Stokes and FitzHugh-Nagumo systems respectively by sparse controls.…”

Section: Sparse Control Of Semilinear Parabolic Equationsmentioning

confidence: 99%

A review on sparse solutions in optimal control of partial differential equations

Rentería

2017

SeMA

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In this paper a review of the results on sparse controls for partial differential equations is presented. There are two different approaches to the sparsity study of control problems. One approach consists of taking functions to control the system, putting in the cost functional a convenient term that promotes the sparsity of the optimal control. A second approach deals with controls that are Borel measures and the norm of the measure is involved in the cost functional. The use of measures as controls allows to obtain optimal controls supported on a zero Lebesgue measure set, which is very interesting for practical implementation. If the state equation is linear, then we can carry out a complete analysis of the control problem with measures. However, if the equation is nonlinear the use of measures to control the system is still an open problem, in general, and the use of functions to control the system seems to be more appropriate.

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

Cited by 26 publications

References 17 publications

A Review of Numerical Analysis for the Discretization of the Velocity Tracking Problem

A Review of Numerical Analysis for the Discretization of the Velocity Tracking Problem

An optimal control problem related to a 3D-chemotaxis-Navier-Stokes model

A review on sparse solutions in optimal control of partial differential equations

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