Abstract
The global stabilization of the Camassa-Holm equation with a distributed feedback control of the form $-(\lambda u-\beta u_{xx}-\lambda[u])$ is investigated. The existence and uniqueness of global strong solutions and global weak solutions to the closed loop control system are obtained. The exponential asymptotical stabilization of weak solutions to the problem is established. Namely, the weak solutions to the problem exponentially uniformly decay to a constant. The main novelty in this paper is that the effects of the coefficients λ and β on the global existence and exponential asymptotical stabilization of solutions are given.
Highlights
1 Introduction This paper is concerned with the distributed feedback control problem for the CamassaHolm equation
For other methods to establish the local well-posedness for the Cauchy problem and global existence of solutions to the Camassa-Holm equation or other shallow water models, the reader is referred to [ – ] and the references therein
Motivated by the work in [, ], we study the global stabilization of problem
Summary
Novruzova and Hagverdiyev [ ] obtained the global existence and uniqueness of strong solutions to Cauchy problem of the Camassa-Holm equation in H (R). For other methods to establish the local well-posedness for the Cauchy problem and global existence of solutions to the Camassa-Holm equation or other shallow water models, the reader is referred to [ – ] and the references therein. Glass [ ] investigated the exact controllability and global asymptotical stabilization of solutions to the Camassa-Holm equation on S by means of a distributed control. Due to the presence of feedback control term, the conserved law which plays an important role in studying the problem disappeared This difficulty has been dealt with by establishing the energy inequality and using the estimates of solutions to the transport equation. The uniqueness of global weak solutions is established with certain assumptions
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