A mixed formulation for the direct approximation of the control of minimal $$L^2$$ L 2 -norm for linear type wave equations

Ĉındea, Nicolae; Münch, Arnaud

doi:10.1007/s10092-014-0116-x

Cited by 29 publications

(88 citation statements)

References 28 publications

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“…The main reason of the present work is to adapt this approach to cover the case ρ := 0 and therefore obtain directly an approximation v h of the control of some minimal weighted L 2 -norm. To do so, we adapt the idea developed in [11] devoted to the wave equation. We also mention [32] where a different space-time variational approach (based on Least-squares principles) is used to approximate null controls for the heat equation.…”

Section:  mentioning

confidence: 99%

“…Consequently, we may conclude that the finite approximation we have used do "pass" the discrete inf-sup test. It is interesting to note that this is in contrast with the situation for the wave equation for which the parameter r have to be adjusted carefully with respect to h; we refer to [11]. Moreover, as it is usual in mixed finite element theory, such a property together with the uniform coercivity of form a ε,r then implies the convergence of the approximation sequence (ϕ h , λ h ) solution of (40).…”

Section: The Discrete Inf-sup Testmentioning

confidence: 99%

“…The resolution amounts to solve a sparse symmetric linear system : the corresponding matrix can be preconditioned if necessary, and may be computed once for all as it does not depend on the initial data to be controlled. Eventually, as discussed in [11], Section 4.3 (but not employed here), the space-time discretization of the domain allows an adaptation of the mesh so as to reduce the computational cost and capture the main features of the solutions. We also emphasize that the higher dimensional case is very similar as it requires C 1 approximation in space.…”

Section: Concluding Remarks and Perspectivesmentioning

confidence: 99%

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A mixed formulation for the direct approximation of L 2-weighted controls for the linear heat equation

Münch

Souza

2015

Adv Comput Math

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This paper deals with the numerical computation of null controls for the linear heat equation. The goal is to compute approximations of controls that drive the solution from a prescribed initial state to zero at a given positive time. In [Fernandez-Cara & Münch, Strong convergence approximations of null controls for the 1D heat equation, 2013], a so-called primal method is described leading to a strongly convergent approximation of distributed control: the controls minimize quadratic weighted functionals involving both the control and the state and are obtained by solving the corresponding optimality conditions. In this work, we adapt the method to approximate the control of minimal square integrable-weighted norm. The optimality conditions of the problem are reformulated as a mixed formulation involving both the state and its adjoint. We prove the well-posedeness of the mixed formulation (in particular the inf-sup condition) then discuss several numerical experiments. The approach covers both the boundary and the inner situation and is valid in any dimension.

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Section:  mentioning

confidence: 99%

Section: The Discrete Inf-sup Testmentioning

confidence: 99%

Section: Concluding Remarks and Perspectivesmentioning

confidence: 99%

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A mixed formulation for the direct approximation of L 2-weighted controls for the linear heat equation

Münch

Souza

2015

Adv Comput Math

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“…In this section, following [10], we introduce a class of controls for both system (1) and the limit one (5) which are smooth for smooth data. Note that, in general, this is not the case for controls with minimal L 2 -norm and the asymptotic analysis below cannot be justified for such controls.…”

Section: Minimal L -Weighted Controlsmentioning

confidence: 99%

“…Definition 2.2 Let η be the weight function introduced at the beginning of Section 2. For any (y 0 , y 1 ) ∈ X we define the minimal L 2 -weighted control v(t) associated to (5) as the function…”

Section: Minimal L 2 -Weighted Controls For the Wave Equationmentioning

confidence: 99%

Singular asymptotic expansion of the exact control for the perturbed wave equation

Castro

Münch

2020

Asymptotic Analysis

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The Petrowsky type equation y ε tt + εy ε xxxx − y ε xx = 0, ε > 0 encountered in linear beams theory is null controllable through Neumann boundary controls. Due to the boundary layer of size of order √ ε occurring at the extremities, these boundary controls get singular as ε goes to 0. Using the matched asymptotic method, we describe the boundary layer of the solution y ε then derive a rigorous second order asymptotic expansion of the control of minimal L 2 −norm, with respect to the parameter ε. In particular, we recover that the leading term of the expansion is a null Dirichlet control for the limit hyperbolic wave equation, in agreement with earlier results due to J-.L. Lions in the eighties. Numerical experiments support the analysis.

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Numerical Estimations of the Cost of Boundary Controls for the Equation y t − εy xx + My x = 0 with Respect to ε

Münch

2018

SEMA SIMAI Springer Series

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A mixed formulation for the direct approximation of the control of minimal $$L^2$$ L 2 -norm for linear type wave equations

Cited by 29 publications

References 28 publications

A mixed formulation for the direct approximation of L 2-weighted controls for the linear heat equation

A mixed formulation for the direct approximation of L 2-weighted controls for the linear heat equation

Singular asymptotic expansion of the exact control for the perturbed wave equation

Numerical Estimations of the Cost of Boundary Controls for the Equation y t − εy xx + My x = 0 with Respect to ε

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