Minimal Orbits and Hamilton-Jacobi Equations

我们将发展周期正定拉格朗日系统的某些方面的数值分析(例如，n-环面上的测地线流)：全局作用最小化半轨道(测地线射线)和弱Kam环面，这提供了对作用泛函的欧拉流行为的一些洞察。将最优控制理论的标准方法应用于这一特殊情况，我们得到了某些Hamilton-Jacobi-Bellman(HJB)方程或双曲型守恒律组的周期(时间和空间)边值问题。在数值格式中，我们考虑这些偏微分方程解的近似解，以构造弱Kam环面及其相关的最小半轨。我们将从离散解的存在性、稳定性、收敛、误差估计等方面对格式进行分析。

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.