Minimal Orbits and Hamilton-Jacobi Equations

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We shall develop the numerical analysis of certain aspects of periodic positive-definite Lagrangian systems (e.g. of geodesic flows on the n-torus): globally action-minimising semi-orbits (geodesic rays) and weak KAM tori, that provide some insight in the behaviour of the Euler flow of the action functional. Adapting a standard approach of optimal control theory to this particular situation, we obtain periodic (in time and space) boundary value problems for certain Hamilton-Jacobi-Bellman (HJB) equations or alternatively hyperbolic systems of conservation laws. The numerical schemes we consider approximate solutions of these PDE problems in order to construct weak KAM tori and associated minimal semi-orbits. We shall analyse the schemes with respect to existence of discrete solutions, stability, convergence, error estimates.