In optimal control problems defined on stratified domains, the dynamics and the running cost may have discontinuities on a finite union of submanifolds of RN. In [8, 5], the corresponding value function is characterized as the unique viscosity solution of a discontinuous Hamilton-Jacobi equation satisfying additional viscosity conditions on the submanifolds. In this paper, we consider a semi-Lagrangian approximation scheme for the previous problem. Relying on a classical stability argument in viscosity solution theory, we prove the convergence of the scheme to the value function. We also present HJSD, a free software we developed for the numerical solution of control problems on stratified domains in two and three dimensions, showing, in various examples, the particular phenomena that can arise with respect to the classical continuous framework.
翻译:在对分层域界定的最佳控制问题中,动态和运行成本在RN的子部件有限结合上可能存在不连续性。在[8、5]中,相应的价值函数被描述为不连续的汉密尔顿-Jacobi等式的独特粘度解决方案,它满足了子部件上的额外粘度条件。在本文中,我们考虑前一个问题的半拉格拉格近似方案。我们依靠在粘度解决方案理论中的传统稳定性论点,证明该办法与值函数的趋同。我们还介绍了HJSD,这是我们为分层域控制问题数字解决方案开发的二、三个层面的免费软件,在各种实例中显示了在传统连续框架方面可能出现的特定现象。