Nash equilibria and systems of Hamilton-Jacobi PDEs

Given a nonzero-sum differential game one can construct a system of Hamilton-Jacobi PDEs. From 1969 it is known that the classical solution of this system determines the value function in the case of Nash equilibrium. However there is no existence theorem for the system of Hamilton-Jacobi PDEs. Moreover the attempts to define the generalized solution of the system Hamilton-Jacobi PDEs were unsuccessful. In the talk we discuss the link between Nash equilibria and system of Hamilton-Jacobi PDEs once more. The presented examples shows that the equilibria constructed by other tools (in particular, using local punishment method) can strongly enhance the equilibria determined by the system of Hamilton-Jacobi PDEs.