Continuity of Optimal Control Costs and Its Application to Weak Kam Theory

Integrability of Hamiltonian systems has been a subject of considerable interest for several decades. One way to understand the dynamics of such systems is to find a family of smooth solutions, called generating functions, to the time-independent Hamilton-Jacobi equation. These generating functions define symplectic transformations which transform the given completely integrable Hamiltonian system to a much simpler one that are easily solvable. On the contrary, if the Hamiltonian system is not completely integrable, then it is natural to ask whether one can solve the Hamilton-Jacobi equation in certain weak sense. This is accomplished in, what is known as, the weak KAM theorem under certain assumptions on the Hamiltonian. More precisely, let L : TM → R be a Lagrangian defined on the tangent bundle TM of a compact manifold M which satisfies the following conditions: