带漂移项的非强制哈密顿量的渐近性质
Asymptotic properties of non-coercive Hamiltonians with drift
AI总结：
本文研究环面上带漂移项的非强制哈密顿量，在强括号生成条件下证明临界值存在、临界方程有粘性解，并用临界常数和临界解刻画演化 H-J 方程的长时间行为，进而应用于周期均匀化。
摘要：
我们研究与环面上仿射系统的最优控制问题相关的哈密顿量。这些哈密顿量不是强制的，并且可能在系统漂移方向上从下方无界。主要假设是向量场上的强括号生成条件。我们首先利用遍历逼近证明哈密顿量临界值的存在性。接着证明临界方程存在一个可能不连续的粘性解。我们证明演化哈密顿-雅可比方程解的长时间行为可以用临界常数和一个临界解来描述。与经典弱 KAM 理论一样，我们找到了哈密顿-雅可比-拉克斯-奥莱尼克半群的一个不动点，尽管该不动点可能不连续。最后，我们将临界值的存在性及其性质应用于稳态和演化 H-J 方程的周期均匀化。
英文摘要：
We consider Hamiltonians associated to optimal control problems for affine systems on the torus. They are not coercive and are possibly unbounded from below in the direction of the drift of the system. The main assumption is the strong bracket generation condition on the vector fields. We first prove the existence of a critical value of the Hamiltonian by means of the ergodic approximation. Next we prove the existence of a possibly discontinuous viscosity solution to the critical equation. We show that the long-time behaviour of solutions to the evolutive Hamilton-Jacobi equation is described in terms of the critical constant and a critical solution. As in the classical weak KAM theory we find a fixed point of the Hamilton-Jacobi-Lax-Oleinik semigroup, although possibly discontinuous. Finally we apply the existence and properties of the critical value to the periodic homogenization of stationary and evolutive H-J equations.
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