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

Abstract

This article is dedicated to improve the controllability results obtained by Cerpa and Pazoto (Commun Contemp Math 13:183–189, 2011) and by Micu et al. (Commun Contemp Math 11(5):779–827, 2009) for a nonlinear coupled system of two Korteweg–de Vries equations posed on a bounded interval. Initially, Micu et al. (2009) 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 Cerpa and Pazoto (2011), 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 Capistrano-Filho et al. (Z Angew Math Phys 67(5):67–109, 2016) 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 linear 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. B Ademir F. Pazoto ademir@im.ufrj.br Roberto A. Capistrano-Filho capistranofilho@dmat.ufpe.br Fernando A. Gallego fgallego@ufrj.br; ferangares@gmail.com 1 Departamento de Matemática, Universidade Federal de Pernambuco, UFPE, Recife, PE CEP 50740-545, Brazil 2 Instituto de Matemática, Universidade Federal do Rio de Janeiro, UFRJ, P.O. Box 68530, Rio de Janeiro, RJ CEP 21945-970, Brazil

DOI: 10.1007/s00498-017-0186-9

Cite this paper

@article{CapistranoFilho2017BoundaryCO, title={Boundary controllability of a nonlinear coupled system of two Korteweg-de Vries equations with critical size restrictions on the spatial domain}, author={Roberto A. Capistrano-Filho and Fernando A. Gallego and Ademir Fernando Pazoto}, journal={MCSS}, year={2017}, volume={29}, pages={6} }