The Existence Of Periodic Solutions For A Class Of Infinite Dimensional Hamiltonian Systems

The study of periodic solutions of Hamiltonian systems has a long and prestigious history,it starts with the invention of calculus.The interest in periodic solutions to Hamil-tonian systems has been unabated for three centuries now.In an oft-quoted sentence[10],Poincare considers it to be " the only opening through which we can force our way into an otherwise impenetrable citadel".In this thesis,We mainly study the existence of periodic solution for the following infinite dimensional Hamiltonian system:This system arises in optimal control of systems governed by partial differential equations.By the theory of Leray-Schauder degree,we study a type of infinite dimensional Hamiltonian system.Under the assumption that the nonlinear term has sublinearity at infinity,the existence should be shown.the results are obtained.