Boundary effects on the controllability of coupled KDV systems

¹ Departamento de Matemática y Estadística, Universidad Nacional de Colombia (UNAL), Cra 27 No. 64-60, 170003, Manizales, Colombia
² Institute of Mathematics, Federal University of Rio de Janeiro, UFRJ, Cidade Universitaria, P.O. Box 68530, CEP 21945-970, Rio de Janeiro, RJ, Brazil
³ Departamento de Matemáticas, Universidad del Valle, Calle 13 No. 100 - 00, Ciudadela Universitaria Meléndez, Cali, Colombia

^* Corresponding author: ivonne.rivas@correounivalle.edu.co

Received: 10 March 2025
Accepted: 16 July 2025

Abstract

We study the exact boundary controllability of a nonlinear coupled system of two Korteweg-de Vries equations on a bounded interval. The model describes the interactions of two weakly nonlinear gravity waves in a stratified fluid. Due to the nature of the system, six boundary conditions are required. However, to study the controllability property, we consider a different combination of the control inputs, with a maximum of four. Firstly, the results are obtained for the linearized system through a classical duality approach and some hidden regularity properties of the boundary terms. This approach reduces the controllability problem to the study of a spectral problem, which is solved by using the Paley–Wiener method introduced by Rosier [ESAIM Control Optim. Cal. Var. 2 (1997) 33–55]. Then, the issue is to establish when a certain quotient of entire functions still turns out to be an entire function. It can be viewed as a problem of factoring an entire function that, depending on the control configuration, leads to the study of a transcendental equation. Finally, by using the contraction mapping theorem, we derive the local controllability for the full system.