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.

No 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.

No abstract

In this paper we study the boundary controllability of the Gear-Grimshaw system posed on a finite domain (0, L), with Neumann boundary conditions:We first prove that the corresponding linearized system around the origin is exactly controllable in (L 2 (0, L)) 2 when h2(t) = g2(t) = 0. In this case, the exact controllability property is derived for any L > 0 with control functions h0, g0 ∈ H − 1 3 (0, T ) and h1, g1 ∈ L 2 (0, T ). If we change the position of the controls and consider h0(t) = h2(t) = 0 (resp. g0(t) = g2(t) = 0) we obtain the result with control functions g0, g2 ∈ H − 1 3 (0, T ) and h1, g1 ∈ L 2 (0, T ) if and only if the length L of the spatial domain (0, L) belongs to a countable set. In all cases the regularity of the controls are sharp in time. If only one control act in the boundary condition, h0(t) = g0(t) = h2(t) = g2(t) = 0 and g1(t) = 0 (resp. h1(t) = 0), the linearized system is proved to be exactly controllable for small values of the length L and large time of control T . Finally, the nonlinear system is shown to be locally exactly controllable via the contraction mapping principle, if the associated linearized systems are exactly controllable.1991 Mathematics Subject Classification. Primary 35Q53; Secondary 37K10, 93B05, 93D15.

The main purpose of this paper is to show the global stabilization and exact controllability properties for a fourth order nonlinear fourth order nonlinear Schrödinger system on a periodic domain T with internal control supported on an arbitrary sub-domain of T. More precisely, by certain properties of propagation of compactness and regularity in Bourgain spaces, for the solutions of the associated linear system, we show that the system is globally exponentially stabilizable. This property together with the local exact controllability ensures that fourth order nonlinear Schrödinger is globally exactly controllable.

In this paper we consider the problem of controlling pointwise, by means of a time dependent Dirac measure supported by a given point, a coupled system of two Korteweg-de Vries equations on the unit circle. More precisely, by means of spectral analysis and Fourier expansion we prove, under general assumptions on the physical parameters of the system, a pointwise observability inequality which leads to the pointwise controllability when we observe two control functions. In addition, with a uniqueness property proved for the linearized system without control, we are also able to show pointwise controllability when only one control function acts internally. In both cases we can find, under some assumptions on the coefficients of the system, the sharp time of the controllability.Date: Version 2019-06-16. 2010 Mathematics Subject Classification. Primary: 93B07, 35Q53 Secondary: 93B52, 93B05.

In this paper we consider the initial boundary value problem of the Korteweg-de Vries equation posed on a finite intervalsubject to the nonhomogeneous boundary conditions,aij∂ j x u(0, t) + bij∂ j x u(L, t) , i = 1, 2, 3, and aij, bij (j, i = 0, 1, 2, 3) are real constants. Under some general assumptions imposed on the coefficients aij, bij, j, i = 0, 1, 2, 3, the IBVPs (0.1)-(0.2) is shown to be locally well-posed in the space H s (0, L) for any s ≥ 0 with φ ∈ H s (0, L) and boundary values hj, j = 1, 2, 3 belonging to some appropriate spaces with optimal regularity.