Proximal aiming in weak KAM theory with nonsmooth Lagrangian

This work extends weak KAM theory to the case of a nonsmooth Lagrangian satisfying a superlinear growth condition. Using the solution of a weak KAM equation that is a stationary Hamilton-Jacobi equation and the proximal aiming method, we construct a family of discontinuous feedback strategies that are nearly optimal for every time interval. This result leads to an analogue of the weak KAM theorem. Additionally, as in classical weak KAM theory, we demonstrate that the effective Hamiltonian (Ma\~{n}\'{e} critical value) can be determined by solving a linear programming problem in the class of probability measures.