Abstract：The stabilization of a nonlinear dispersive shallow water wave equation, the Camassa - Holm equation ut -uxxt ＋ 3uux = 2uxuxx ＋ uuxxx is studied. By adding a feedback control term -k （u -uxx - [ u ] ） （ k 〉 0） on this equation, the solution to the controlled closed-loop system with the period boundary conditions u （0, t） = u （ 1, t） = ux （0, t） = ux （ 1, t） = 0, is proved to decay exponentially to a constant in H^1-norm sense. [ u ] denotes the total volume of water. The action of the ＂feedback＂ term- k（ u -uxx - [ u] ） consists of balancing the level of water, conserving at the same time its total volume of water. The periodic boundary conditions correspond to a circular movement. Kato＇s theory guarantees the local existence of solution to the closed-loop system in space H^2 p. By the H^1 global exponential stability, the solution to the closed-loop system is shown to exist globally in space H^1 p. The results may contribute to further theoretical research or engineering applications.
2008, Vol. 29 
