Boundary controllability of a nonlinear coupled system of two Korteweg–de Vries equations with critical size restrictions on the spatial domain

Abstract

Abstract. This article is dedicated to improve the controllability results obtained by Cerpa et al. in [4] and by Micu et al. in [11] for a nonlinear coupled system of two Korteweg-de Vries (KdV) equations posed on a bounded interval. Initially, in [11], the authors proved that the nonlinear system is exactly controllable by using four boundary controls without any restriction on the length L of the interval. Later on, in [4], two boundary controls were considered to prove that the same system is exactly controllable for small values of the length L and large time of control T . Here, we use the ideas contained in [2] to prove that, with another configuration of four controls, it is possible to prove the existence of the so-called critical length phenomenon for the nonlinear system, i. e., whether the system is controllable depends on the length of the spatial domain. In addition, when we consider only one control input, the boundary controllability still holds for suitable values of the length L and time of control T . In both cases, the control spaces are sharp due a technical lemma which reveals a hidden regularity for the solution of the adjoint system.

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

“…To our knowledge, this problem has not been addressed in the literature and the existing developments do not allow to give an immediate answer to it. Indeed, the controllability problem for system (1.1) was first solved when the control acts through the boundary conditions [8,9,17,35]. The results are obtained combining the analysis of the linearized system and the Banach's fixed-point theorem.…”

“…Later on, in [35], the authors proved the local exact boundary controllability property for the nonlinear system (1.1) posed on a bounded interval. Their result was improved in [9], [17] and in [8], where the same boundary control problem was addressed with a different set of boundary conditions. In all those works, except in [17], the results were proved by applying the duality approach and some ideas introduced by Rosier in [36].…”

Pointwise control of the linearized Gear-Grimshaw system

“…Their result was improved by Cerpa and Pazoto [8] and by Capistrano-Filho et al [6]. By considering a different set of boundary conditions, the same boundary control problem was also addressed by the authors in [7]. We note that, the results mentioned above were first obtained for the corresponding linearized systems by applying the Hilbert Uniqueness Method (HUM) due to J.-L. Lions [24], combined with some ideas introduced by Rosier in [30].…”