A representational formula for variational solutions to Hamilton-Jacobi equations

Abstract

For Cauchy problems given by Hamilton-Jacobi evolutive type equations, we consider the variational solution proposed by Chaperon, Sikorav and Viterbo. This is a weak, Lipschitz solution constructed via a minimax procedure from the generating function quadratic at infinity of the Lagrangian manifold associated with the Cauchy problem. We state and prove a representational formula for the variational solution. This formula requires a condition on the nature of the minimax critical value of the generating function, but makes no assumption about the convexity or concavity of the Hamiltonian. We show that it generalises the well-known formula which applies when the Hamiltonian is convex or concave in the momentum variable. We then prove that the required conditions of the formula are satisfied by the non-convex Hamiltonian arising from the control-affine $H_{\infty }$ problem. Given results in the literature that the variational solution to this problem is equivalent to the lower value of the associated differential game, we therefore obtain a representational formula for this lower value.