Null controllability of a penalized Stokes problem in dimension two with one scalar control

Bárcena-Petisco, Jon Asier

doi:10.3233/asy-191550

Cited by 5 publications

(10 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…but the uniqueness is only valid in the 2D case. • Theorem 1.1 is a null controllability result, uniform with respect to the parameter ε > 0 because of the estimate (3). So, by letting ε → 0 in (1), we recover the results from [7] for less regular initial data but for more restrictive assumptions on Ω, ω.…”

Section: Introductionmentioning

confidence: 56%

“…We recall that, as proved in [3], Hypothesis 1 is satisfied by any strictly convex smooth domain Ω. Moreover, according to [3,Lemma 1.3], for any smooth domain Ω, one can find a rotation (i.e. a linear application of the type U θ (x, y) = ((cos θ)x − (sin θ)y, (sin θ)x + (cos θ)y)) that maps Ω to a domain Ω satisfying Hypothesis 1.…”

Section: Introductionmentioning

confidence: 66%

“…Example 1. We recall that, as proved in [3], Hypothesis 1 is satisfied by any strictly convex smooth domain Ω. Moreover, according to [3,Lemma 1.3], for any smooth domain Ω, one can find a rotation (i.e.…”

Section: Introductionmentioning

confidence: 87%

See 2 more Smart Citations

Local null controllability of the penalized Boussinesq system with a reduced number of controls

Bárcena-Petisco

Balc’h

2022

MCRF

Self Cite

View full text Add to dashboard Cite

<p style='text-indent:20px;'>In this paper we consider the Boussinesq system with homogeneous Dirichlet boundary conditions, defined in a regular domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset\mathbb R^N $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M2">\begin{document}$ N = 2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ N = 3 $\end{document}</tex-math></inline-formula>. The incompressibility condition of the fluid is replaced by its approximation by penalization with a small parameter <inline-formula><tex-math id="M4">\begin{document}$ \varepsilon > 0 $\end{document}</tex-math></inline-formula>. We prove that our system is locally null controllable using a control with a restricted number of components, localized in an open set <inline-formula><tex-math id="M5">\begin{document}$ \omega $\end{document}</tex-math></inline-formula> contained in <inline-formula><tex-math id="M6">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula>. We also show that the control cost is bounded uniformly with respect to <inline-formula><tex-math id="M7">\begin{document}$ \varepsilon \rightarrow 0 $\end{document}</tex-math></inline-formula>. The proof is based on a linearization argument. The null controllability of the linearized system is obtained by proving a new Carleman estimate for the adjoint system. This inequality is derived by exploiting the coercivity of some second order differential operator involving crossed derivatives.</p>

show abstract

Section: Introductionmentioning

confidence: 56%

Section: Introductionmentioning

confidence: 66%

See 1 more Smart Citation

Local null controllability of the penalized Boussinesq system with a reduced number of controls

Bárcena-Petisco

Balc’h

2022

MCRF

Self Cite

View full text Add to dashboard Cite

show abstract

“…From a controllability point of view, the null-controllability of the Stokes system, respectively Navier-Stokes, is established in [19], respectively in [1], with as many controls as equations. More recently, [6] proves the null-controllability of the Stokes system in 2D with only one control, this result having been extended in [7] for the (nonlinear)…”

Section: Introductionmentioning

confidence: 81%

A penalty approach to the infinite horizon LQR optimal control problem for the linearized Boussinesq system

Balc’h

Tucsnak

2021

ESAIM: COCV

View full text Add to dashboard Cite

In this paper, we consider the infinite time horizon LQR optimal control problem for the linearized Boussinesq system. The goal is to justify the approximation by penalization of the free divergence condition in this context. We establish convergence results for optimal controls, optimal solutions and Riccati operators when the penalization parameter goes to zero. These results are obtained under two different assumptions. The first one treats the linearization arround a sufficiently small stationary state and an arbitrary control operator (possibly of finite rank), while the second one does no longer require the smallness of the stationary state but requires to consider controls distributed in a subdomain and depending on the space variable.

show abstract

“…For instance, there are a lot of results about Navier-Stokes-like systems, such as [7,13,14,18,19,24,30,33]. It has also been studied, for example, the controllability in cascade-like systems [29,15], the null controllability in the context of linear thermoelasticity [31], the controllability to trajectories in phase-field models [3], the existence of insensitizing controls for the heat equation [20], and the controllability in reaction-diffusion systems [2].…”

Section: Historical Backgroundmentioning

confidence: 99%

Local null controllability of a model system for strong interaction between internal solitary waves

Bárcena-Petisco

Guerrero

Pazoto

2021

Commun. Contemp. Math.

Self Cite

View full text Add to dashboard Cite

In this paper, we prove the local null controllability property for a nonlinear coupled system of two Korteweg–de Vries equations posed on a bounded interval and with a source term decaying exponentially on [Formula: see text]. The system was introduced by Gear and Grimshaw to model the interactions of two-dimensional, long, internal gravity waves propagation in a stratified fluid. We address the controllability problem by means of a control supported on an interior open subset of the domain and acting on one equation only. The proof consists mainly on proving the controllability of the linearized system, which is done by getting a Carleman estimate for the adjoint system. While doing the Carleman, we improve the techniques for dealing with the fact that the solutions of dispersive and parabolic equations with a source term in [Formula: see text] have a limited regularity. A local inversion theorem is applied to get the result for the nonlinear system.

show abstract

Null controllability of a penalized Stokes problem in dimension two with one scalar control

Cited by 5 publications

References 25 publications

Local null controllability of the penalized Boussinesq system with a reduced number of controls

Local null controllability of the penalized Boussinesq system with a reduced number of controls

A penalty approach to the infinite horizon LQR optimal control problem for the linearized Boussinesq system

Local null controllability of a model system for strong interaction between internal solitary waves

Contact Info

Product

Resources

About