Internal controllability of the korteweg–de vries equation on a bounded domain

Abstract

Abstract. This paper is concerned with the control properties of the Korteweg-de Vries (KdV) equation posed on a bounded interval with a distributed control. When the control region is an arbitrary open subdomain, we prove the null controllability of the KdV equation by means of a new Carleman inequality. As a consequence, we obtain a regional controllability result, the state function being controlled on the left part of the complement of the control region. Finally, when the control region is a neighborhood of the right endpoint, an exact controllability result in a weighted L 2 -space is also established.

Null Controllability of a Coupled System of Two Korteweg-de Vries Equations from the Left Dirichlet Boundary Conditions

“…Applying Theorem 2.1, we see that ((ξ, ζ )) is nonempty for all (ξ, ζ ) ∈ B. The fact that ((ξ, ζ )) is closed in F for every (ξ, ζ ) ∈ B and that is upper semicontinuous can be obtained following the methods developed in [16] with minor changes.…”

“…First, we review an estimate for the following system: The existence of such ψ can be found in [16].…”

“…The following part is close to [16,18]; therefore, we just sketch it. Claim that for any (η T , w T ) ∈ (L 2 (−L, L)) 2 and M i , N i ∈ X (i = 1, 2), the solution (η, w) of Eq.…”

“…1.4 and motivated by [16,18], it is natural to consider the null controllability of system (1.1)-(1.3).…”

Local exact controllability to the trajectories of the Korteweg–de Vries–Burgers equation on a bounded domain with mixed boundary conditions

“…Other related results can be found in [19] and [32]. Concerning the internal controllability for the KdV equation on a bounded domain with homogeneous Dirichlet boundary conditions, the most recent work was done by Capistrano-Filho et al in [7], where the authors obtained some controllability results using an approach based on Carleman estimates and weighted Sobolev spaces.…”