Exact boundary controllability of a nonlinear KdV equation with critical lengths

Coron, Jean‐Michel; Crépeau, Emmanuelle

doi:10.4171/jems/13

Cited by 133 publications

(171 citation statements)

References 16 publications

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“…In this case, that is when a ≡ 0, it is important to observe that Theorem 3.2 holds if the following unique continuation principle can be proved: As far as we know, the above unique continuation principle remains to be done. Coron and Crépeau [6] studied the boundary controllability of the nonlinear KdV equation with Dirichlet boundary condition on an interval with critical length and they proved that the nonlinear term gives the local controllability around the origin.…”

Section: Final Remarksmentioning

confidence: 99%

Stabilization of the Kawahara equation with localized damping

Vasconcellos

Silva

2009

ESAIM: COCV

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Abstract.We study the stabilization of global solutions of the Kawahara (K) equation in a bounded interval, under the effect of a localized damping mechanism. The Kawahara equation is a model for small amplitude long waves. Using multiplier techniques and compactness arguments we prove the exponential decay of the solutions of the (K) model. The proof requires of a unique continuation theorem and the smoothing effect of the (K) equation on the real line, which are proved in this work.Mathematics Subject Classification. 35Q35, 35B40, 35Q53.

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Section: Final Remarksmentioning

confidence: 99%

Stabilization of the Kawahara equation with localized damping

Vasconcellos

Silva

2009

ESAIM: COCV

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“…To get the controllability of the KdV equations, "Power Series Expansion" method was introduced in [1,3,13], which turned out to be a classical example of getting controllability by using nonlinear terms. The stabilization problem is even more interesting, as we need to investigate a closed-loop system which involves more difficulties (even for the well-posedness).…”

Section: Introductionmentioning

confidence: 99%

Small-time local stabilization for a Korteweg–de Vries equation

Xiang

2018

Systems & Control Letters

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This paper focuses on the (local) small-time stabilization of a Korteweg-de Vries equation on bounded interval, thanks to a time-varying Dirichlet feedback law on the left boundary. Recently, backstepping approach has been successfully used to prove the null controllability of the corresponding linearized system, instead of Carleman inequalities. We use the "adding an integrator" technique to gain regularity on boundary control term which clears the difficulty from getting stabilization in small-time.

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“…By a linearization argument, a local controllability result for the semilinear scalar equation was also proved in [22]. Later on, Zhang in [28] proved that using three controls, acting on all the boundary conditions, controllability holds for all values of L. More recently, Crépeau and Coron [5] and Cerpa [3] proved that, for some critical values of the length L, the nonlinear scalar model is controllable (see also [4]). When the damping function a = a(x) is active simultaneously in a neighborhood of both extremes of the interval (0, L) the problem was addressed in [16] for the scalar KdV equation…”

Section: ||U (· T)||mentioning

confidence: 91%

Uniform stabilization of a class of coupled systems of KdV equations with localized damping

2011

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Abstract. We study the stabilization of solutions of a coupled system of Kortewegde Vries (KdV) equations in a bounded interval under the effect of a localized damping term. We use multiplier techniques combined with the so-called "compactness-uniqueness argument". The problem is then reduced to proving a unique continuation property (UCP) for weak solutions. The exponential decay of solutions was previously obtained in Bisognin, Bisognin, and Menzala (2003) when the damping was effective simultaneously in neighborhoods of both extremes of the bounded interval. In this work we address the general case using a different approach to obtain the UCP and stabilize the system.

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Exact boundary controllability of a nonlinear KdV equation with critical lengths

Cited by 133 publications

References 16 publications

Stabilization of the Kawahara equation with localized damping

Stabilization of the Kawahara equation with localized damping

Small-time local stabilization for a Korteweg–de Vries equation

Uniform stabilization of a class of coupled systems of KdV equations with localized damping

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