我们主要研究Hamitlon-Jacobi 方程粘性解的奇点动力学，以及它在Mather理论及弱KAM 理论中的应用。在大于Mane 临界值的能量面上，弱KAM 不存在孤立奇点。奇点的动力学及其稳定性与Aubry 类的拓扑结构、共轭点的产生与消失，粘性解以及障碍函数的正则性，局部极小轨道的构造有着密切的联系。我们利用最优控制理论与Hamilton 动力学相结合的方法，拟解决以下问题：1、找出Hamilton-Jacobi 方程粘性解奇点沿广义特征线的传播条件；2、研究奇点传播与障碍函数关于上同调类具备某种正则性的关系；3、给出带控制项的 Hamilton-Jacobi 方程粘性解的变分结构。

We will study the singularity dynamics and its stability problem of the viscosity solutions of the H-J equations with its applications to the Mather theory and the weak KAM theory, and its applications to the Hamiltonian dynamics.We know now that the singularity of the barrier functions will propagate on the supercritical energy surface as well as that of the weak KAM solutions. This research area will be great application of the optimal control theory to some deep results in Hamiltonian dynamics. We hope to solve the following problems: .1. We intend to find an analytic setting for the propagation of singularities on viscosity solutions along the generalized characteristics; .2. We want to study the relationship between the regulation of the barrier functions and the propagation of the singular points; .3. We shall give the variational constructions on the viscosity solutions of the H-J equations with the optimal control items.