Stabilization of the Gear–Grimshaw system on a periodic domain

Abstract

Abstract. This paper is devoted to the study of a nonlinear coupled system of two Korteweg-de Vries equations in a periodic domain under the effect of an internal damping term. The system was introduced Gear and Grimshaw to model the interactions of two-dimensional, long, internal gravity waves propagation in a stratified fluid. Designing a time-varying feedback law and using a Lyapunov approach we establish the exponential stability of the solutions in Sobolev spaces of any positive integral order.

Stabilization of the Gear–Grimshaw System with Weak Damping

“…However, the stabilization results for system (1.1)-(1.2) was first obtained in [4], when the authors considered the system in a periodic domain. Recently, Capistrano-Filho et al [3] proved a result which extends the result proved by Dávila [4], which one was proved only for s ≤ 2. More precisely, in [3], they showed that for any fixed integer s ≥ 3, the solutions are exponentially stable in the Sobolev spaces H s p (0, 1) := {u ∈ H s (0, 1) : ∂ n x u(0) = ∂ n x u(1), n = 0, .…”

Pointwise control of the linearized Gear-Grimshaw system

“…Theorem 1.1. For almost all quadruples (a, c, d, r) ∈ (0, ∞) 4 the following property holds. For any fixed…”

“…The internal stabilization problem has also been addressed (see, for instance, [4,9,29] and the references therein). Although controllability and stabilization problems are closely related, one may expect that some of the available results will have some counterparts in the context of the control problem, but this issue is open.…”

“…Although controllability and stabilization problems are closely related, one may expect that some of the available results will have some counterparts in the context of the control problem, but this issue is open. Particularly, when the model is posed on a periodic domain, Capistrano-Filho et al [4], designed a time-varying feedback law and established the exponential stability of the solutions in Sobolev spaces of any positive integral order by using a Lyapunov approach. This extends an earlier theorem of Dávila [9] also obtained in H s (T), for s ≤ 2.…”

“…Such conservation lead to the construction of a suitable Lyapunov function that gives the exponential decay of the solutions. In [4], the use of the Lyapunov approach was possible thanks to the results established by Dávila and Chavez [10]. They proved that, under suitable conditions on the coefficients of the system, the system also has an infinite family of conservation laws.…”