Optimal Control Of The Viscous Peaked Solitary Equations

According to the optimal control theories about variational inequality and distributed parameter system, we have studied a typical optimal control problem for some viscous peaked solitary equations, the form is as follows:where J:W(V)×L²(Q₀)→R and e:W (V)×L² (Q₀)→L²(V)×H are abundant smooth function. W(V), L²(Q₀) and L²(V) are both Hilbert space.W(V) and L² (Q₀) are state space and control space respectively.This paper studies the problems for optimal control of the viscous generalized Camassa-Holm equation and the viscous fifth order shallow water equation. At first, the existence and uniqueness of weak solution in the interval to the viscous generalized Camassa-Holm, the viscous fifth order shallow water equation are proved using Galerkin method. Then, according to the optimal control theories about variational inequality and distributed 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 viscous generalized Camassa-Holm equation and the viscous fifth order shallow water equation under boundary condition are given in L² space, and the existences of optimal solution are proved in theory.