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

Abstract

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.

Control Problems for the Navier–Stokes System with Nonlocal Spatial Terms

“…Remark 1.5. As explained above, if we remove the nonlocal spatial terms of our system, the above result improves the corresponding result in [6] since in this previous work, one need g (1) ∈ L 2 (0, T ; H 1 (Ω)) with our approach we manage to remove this condition. A tool that we introduce here to deal with these regularity problems is an adequate decomposition of the solution of the adjoint system.…”

“…We are going to use Proposition 2.1 to estimate φ (1) and φ (2) and we show a Carleman estimate on φ (3) . In order to state this estimate, we introduce to quantities associated with φ (3) :…”

“…In many articles devoted to the controllability of the Navier-Stokes system, the solution of the adjoint system is decomposed into two functions and here we show how to generalize such a decomposition by splitting the solution in several functions which allows to work with a regular solution for the Carleman estimate. A similar decomposition is already proposed in [1] but they need to modify the Carleman weights so that they are regular at t = 0. Such a strategy can not be applied in the context of insensitizing controls since we have to work with a forward system and a backward system.…”

“…Such a strategy can not be applied in the context of insensitizing controls since we have to work with a forward system and a backward system. We thus need to consider another decomposition as in [1].…”

Impulse null approximate controllability for heat equation with dynamic boundary conditions

“…Absorbing this term turns out to be a main difficulty that will be overcome with help of some parabolic regularity estimates with explicit constants in terms of the final time T (see Proposition 3). Such regularity estimates can be very useful while doing Carleman estimates [5]. It is worth mentioning the paper [10] which is the first to deal with a Carleman inequality when a time derivative appears in the boundary condition (but without a boundary diffusion).…”