Abstract
The nonhomogeneous initial boundary value problem for the two-component Camassa-Holm equation, which describes a generalized formulation for the shallow water wave equation, on an interval is investigated. A local in time existence theorem and a uniqueness result are achieved. Next by using the fixed-point technique, a result on the global asymptotic stabilization problem by means of a boundary feedback law is considered.
Highlights
In this paper, we are concerned with the initial boundary value problem and the asymptotic stabilization of the two-component Camassa-Holm equation on a compact interval by means of a stationary feedback law acting on the boundary
Our aim of this paper is to prove the existence of the initial boundary value problem and the asymptotic stabilization of the two-component Camassa-Holm equation on a compact interval by acting on the boundary feedback law, precisely, 1 the exact controllability problem: given two states u0, ρ0 and u1, ρ1 and a time T > 0, can one find a certain function v t such that the solution to 1.1 satisfies u T u1, ρ T ρ1? and 2 the stabilizability problem: can one find a stationary feedback law v x, such that for any state u0, ρ0 a solution pair u t, ρ t to closed-loop system is global?
We conclude with a classical comparison principle for ODES
Summary
We are concerned with the initial boundary value problem and the asymptotic stabilization of the two-component Camassa-Holm equation on a compact interval by means of a stationary feedback law acting on the boundary. As far as the initial boundary value problem and the asymptotic stabilization of the two-component Camassa-Holm equation on a compact interval are concerned, there are seldom results yet, to the authors’ knowledge. Our aim of this paper is to prove the existence of the initial boundary value problem and the asymptotic stabilization of the two-component Camassa-Holm equation on a compact interval by acting on the boundary feedback law, precisely,. The following lemma, see 33 , will play an important role in proving the local time existence theorem and of a uniqueness result of the initial boundary value problem.
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