Control of a Boussinesq system of KdV–KdV type on a bounded interval

Abstract

We consider a Boussinesq system of KdV-KdV type introduced by J. Bona, M. Chen and J.-C. Saut as a model for the motion of small amplitude long waves on the surface of an ideal fluid. This system of two equations can describe the propagation of waves in both directions, while the single KdV equation is limited to unidirectional waves. We are concerned here with the exact controllability of the Boussinesq system by using some boundary controls. By reducing the controllability problem to a spectral problem which is solved by using the Paley-Wiener method introduced by the third author for KdV, we determine explicitly all the critical lengths for which the exact controllability fails for the linearized system, and give a complete picture of the controllability results with one or two boundary controls of Dirichlet or Neumann type. The extension of the exact controllability to the full Boussinesq system is derived in the energy space in the case of a control of Neumann type. It is obtained by incorporating a boundary feedback in the control in order to ensure a global Kato smoothing effect.2000 Mathematics Subject Classification. Primary: 35Q53, Secondary: 37K10, 93B05, 93D15.

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

“…Before closing this section, we mention the work [12], in which the boundary controllability problem was addressed for a Boussinesq system of KdV type.…”

“…in Q, u(⋅, 0) = u(⋅, L) = u x (⋅, L) = 0, on (0, T ), v(⋅, 0) = v(⋅, L) = v x (⋅, L) = 0, on (0, T ), u(0, ⋅) = u 0 , v(0, ⋅) = v 0 , on (0, L),(4 12). …”

Decay of solutions for a dissipative higher-order Boussinesq system on a periodic domain

“…Then, as an application of the newly established exact controllability results, some simple feedback controls are constructed for some particular choice of the parameters, such that the resulting closed-loop systems are exponentially stable. Later on, the stabilization problem was studied in [8,16] for Boussinesq system of KdV-KdV type (b = d = 0) posed on a bounded interval. In any case, depending on the values of its parameters, system (3) couples two equations that may be of KdV-KdV or BBM-BBM types.…”

Asymptotic behavior of Boussinesq system of KdV–KdV type

“…To prove it, we use the classical approach given by the Riesz representation Theorem to obtain a solution by transposition, see [5,6] for more details. Our analysis on the case of regular data (Theorem 1.1) suggests that is possible to obtain the rapid exponential stabilization for the linear system (5.5) for less regularity of the initial data whenever the linear system is well-posedness in some sense on X 0 .…”

“…satisfies η(•, T ) = η T and w(•, T ) = w T ? More recently, in [6] (see also [5]), the exact boundary controllability of the linear system Boussinesq of KdV-KdV type was studied. It was discovered that whether the associated linear system is exactly controllable or not depends on the length of the spatial domain.…”