论文标题
Aubry套件用于亚riemannian控制系统
Aubry set for sub-Riemannian control systems
论文作者
论文摘要
在论文中[P. Cannarsa，C。Mendico，《欧几里得空间上汉密尔顿 - 雅各布贝尔曼方程的渐近分析》，（2021）arxiv]，随着时间范围进入最佳控制问题的平均值函数的无限函数，我们证明了极限的存在。对于经典的Tonelli情况，这种限制称为问题的临界常数。在亚瑞曼次控制系统的特殊情况下，我们还证明了存在关键解决方案的存在，即对与这种常数相关的汉密尔顿 - 雅各比方程的连续解决方案，这也与其Lax-Oleinik Evolution相吻合。在这里，我们将注意力集中在次曼尼亚案例上，为临界常数提供了差异表示公式，该临界常数使用了适应的封闭措施概念。我们可以使用这样的公式，定义和研究Aubry集合。首先，我们研究了此类W.R.T.的动力学和拓扑特性。适当的一类最小化拉格朗日动作的轨迹。然后，我们表明，汉密尔顿 - 雅各比方程的关键解决方案在Aubry集合上具有水平分化，并满足经典意义上的方程。
In the paper [P. Cannarsa, C. Mendico, Asymptotic analysis for Hamilton-Jacobi- Bellman equations on Euclidean space, (2021) Arxiv], we proved the existence of the limit as the time horizon goes to infinity of the averaged value function of an optimal control problem. For the classical Tonelli case such a limit is called the critical constant of the problem. In the special case of sub-Riemannian control systems, we also proved the existence of a critical solution, that is, a continuous solution to the Hamilton-Jacobi equation associated with such a constant, which also coincides with its Lax-Oleinik evolution. Here, we focus our attention on the sub- Riemannian case providing a variational representation formula for the critical constant which uses an adapted notion of closed measures. Having such a formula at our disposal, we define and study the Aubry set. First, we investigate dynamical and topological properties of such a set w.r.t. a suitable class of minimizing trajectories of the Lagrangian action. Then, we show that critical solutions to the Hamilton-Jacobi equation are horizontally differentiable and satisfy the equation in classical sense on the Aubry set.
