Optimal Control Of A Class Of Nonlinear Viscous Dispersive Wave Equations

Optimal control is an important part of 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 Burgers equation,Kdv equation,Kdvb equation and K-S equation. In this paper, we have studied a class of nonlinear viscous dispersive wave equations:the viscous Fornberg-Whitham equation and the viscous dispersive wave equation. Fornberg-Whitham equation is not integrable, and its kink-like wave solutions and antikink-like wave solutions have recently been studied. The dispersive wave equation is a class of nonlinear wave equations.According to the optimal control theories about variational inequality and distributed parameter system, we have studied a typical optimal control problem for above nonlinear viscous dispersive wave equations. First, the existence and uniqueness of weak solution in the interval to above two equations are proved using Galerkin method. Then,according to the optimal control theories about variational inequality anddistributed parameter system,it is proved that in the special Hilbert space,the norm of solution to these two equations are related to the control item and initial value. Finally, the optimal control of the Fornberg-Whitham equation and the dispersive wave equation under boundary condition are given in L2 space, and the existences of optimal solution are proved in theory.