GLOBAL CONTROLLABILITY OF A NONLINEAR KORTEWEG–DE VRIES EQUATION

2009 ◽  
Vol 11 (03) ◽  
pp. 495-521 ◽  
Author(s):  
MARIANNE CHAPOULY

We are interested in both the global exact controllability to the trajectories and in the global exact controllability of a nonlinear Korteweg–de Vries equation in a bounded interval. The local exact controllability to the trajectories by means of one boundary control, namely the boundary value at the left endpoint, has already been proved independently by Rosier, and Glass and Guerrero. We first introduce here two more controls: the boundary value at the right endpoint and the right member of the equation, assumed to be x-independent. Then, we prove that, thanks to these three controls, one has the global exact controllability to the trajectories, for any positive time T. Finally, we introduce a fourth control on the first derivative at the right endpoint, and we get the global exact controllability, for any positive time T.

2019 ◽  
Vol 25 ◽  
pp. 76
Author(s):  
Chaohua Jia ◽  
Wei Guo ◽  
Diao Luo

This paper is concerned with the parameter estimation and boundary feedback stabilization for the linear Korteweg-de Vries equation posed on a finite interval with the boundary observation at the right end and the non-collocated control at the left end. The boundary observation suffers from some unknown disturbance. An adaptive observer is designed and the adaptive laws of the parameters are obtained by the Lyapunov method. The resulted closed-loop system is proved to be well-posed and asymptotically stable in case that the length of the interval is not critical. Moreover, it is shown that the estimated parameter converges to the unknown parameter. As a by-product, a hidden regularity result is proved.


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