Boundary Control Of A Class Of Nonlinear Dissipative Wave Equations

In this paper,we study mainly the problem of the boundary control of the nonlinear dissipative wave equations,and the exponential stability estimate under the given boundary condition.Boundary control is an important part of the modern control theory,which has been emphasized in the control theory field and has been extensively studied and developed.Recently many researches have been done on Burger's equation,KdV equation,KdVB equation and K-S equation.We study a class of nonlinear dissipative wave equations:the viscous Fornberg-Whitham equation,the viscous generalized Camassa-Holm equation and the modified viscous b-family equation.Fornberg-Whitham equation is not integrable, and its kink-like wave solutions and antikink-like wave solutions have recently been studied.Camassa and Holm derived a new completely integrable dispersive wave equation for water waves by Hamiltonian method,namely Camassa-Holm equation (i.e.,CH equation).It has bi-hamiltonian structure and infinite sequence of conserved equatities,and it is completely integrable.The b-family equation is a class of nonlinear wave equations.For example,it is Camassa-Holm equation as b = 2.We mainly studied the boundary control of the above three equations in this paper.At first,the existence and uniqueness of weak solution in the interval to the viscous Fornberg-Whitham equation,the viscous generalized Camassa-Holm equation,and the modified viscous b-family equation are proved by using Galerkin method.Then,according to the theory of functional analysis,energy integration, partial differential equations and Sobolev space and a number of theories related to inequality estimate,it is proved that the solution has the global exponential stability in L~2,H~1,H~2,H~3 under the given boundary condition.