Optimal Control For Some Solitary Wave Equations

According to the optimal control theory about variational inequality and distributed parameter system,we have studied a typical optimal control problem about some solitary wave equations, the form is as follows:min J(y,u) ande(y,u) = 0here J: Y×U → R,e: Y×U → Z~*;is abundant smooth function, Y,U and Z are all Hilbert space, Z~* is the dual of Z,set Y,U is state space and control space.On the basis of presented results ,the paper study some optimal control quantity about Burgers equation include giving the F-derivatives of J and e, proving the regular point condition, the First-order necessary optimality conditions and the Second-order sufficient optimality condition. Further, investigating optimal control problem of KdV-Burgers equation, according to the optimal control theory about variational inequality and distributed parameter system, choosing suitable performance index J(y, u),then it is proofed that the solution is related with control item and original value. The optimal control of KdV-Burgers equation under the Dirichlet boundary condition is given and the existence of optimal solution is proofed. At the same time, The paper study the optimal control for nonlinear strength Burgers equation under Neumann boundary condition. According to Galerkin method, it is proofed that the solution exist in a short interval time. Choosing suitable performance index J(y,u) , the optimal control of nonlinear strength Burgers equation under the Neumann boundary condition is given and the existence of optimal solution is proofed.