弱KAM理论与Mather理论的应用研究

Mather theory has an important assumption that the Hamilton function is time-periodic. Weak KAM theory has similar results just for periodic Hamiltonian systems. The project plans using the core concept Lax-Oleinik semigroup to build similar weak KAM theory for quasi-periodic Hamiltonian systems which is very important for control， optimization and other fields. The weak KAM theory of quasi-periodic Hamiltonian systems is an important foundation of the weak KAM theory of non-periodic systems..The weak KAM theory is a bridge from Mather Theory to the theory of Hamilton-Jacobi equations' viscous solution. They have got many same conclusions from different methods. There are important intrinsic links between the three theories. This topic plans to combine Mather theory's variation method and PDE methods of viscosity solution theory to study the intrinsic links between the speed of the semigroup convergence and the traces' dynamical properties in the minimal invariant sets, and the numerical relationship between the semigroup convergence speed and the viscosity solution's Holder index.

Mather理论的一个重要假设是Hamilton方程对时间的周期性.弱KAM理论也只是对周期Hamilton系统才有类似的结果。本项目计划从引入Lax-Oleinik半群这一核心概念入手，对控制、最优化等领域中非常重要的拟周期Hamilton系统建立类似的弱KAM理论。拟周期系统也是从周期系统的弱KAM理论到非周期系统弱KAM理论的重要基础。.弱KAM理论是沟通Mather理论与Hamilton-Jacobi方程粘性解理论的桥梁。三者虽然所应用的方法不同，但有许多一致的结论，三者之间存在有重要的内在联系。本课题计划结合Mather理论的变分法及粘性解理论的PDE方法，研究半群收敛速度与"极小不变集上轨道的动力学性质"之间的内在联系，以及"半群收敛速度"与"粘性解对平均作用量的Holder指数"间的数量关系。

Mather 理论与弱KAM理论的一个重要假设是 Hamilton 函数对时间的周期性. 这一要求并不自然, 很多物理和控制领域中的问题不满足这一要求. 本项目从引入 Lax-Oleinik 半群这一核心概念入手，对控制、最优化等领域中非常重要的拟周期 Hamilton 系统弱KAM解的长时间性质做了研究。研究取得了很好的成果, 结果不但对逆周期系统成立, 而且对非周期Hamiltonian也都成立..对时间非周期的Hamilton系统, 我们定义了一个新的范数, 证明了粘性解在这一新的范数下一致收敛, 并且其相图也有相应的收敛性. 这一结果与自治的情况一样好..除此之外, 我们还研究了Hamilton系统的周期解及其稳定性问题, 找到了一些稳定周期解和一些不稳定周期解.