A study on the Hamilton-Jacobi equation arising from nonlinear control theory

首先，在Hamilt on-Jacobi方程的背景下概述了一阶偏微分方程组的理论。根据该理论，要求解Hamilton-Jacobi方程，只需解一个常微分方程组和/或积分一个完全可积的Pfaffian系统即可。其次，研究了一种在控制理论中占有重要地位的特殊解--镇定解。作为Hamilton-Jacobi方程理论的一部分，评述了Arimoto和Potter的著名理论，即存在稳定解的充要条件。在控制理论中，我们感兴趣的是获得尽可能多的解。要做到这一点，将需要更多的关于哈密顿-雅可比方程结构的信息，如一个例子所示。最后，介绍了辛变换的母函数理论，这将在后面的小节中发挥核心作用。在给定Hamilton-Jacobi方程的解的情况下，利用辛变换的母函数推导出确定其他解的辅助方程。还可以看到，辅助方程甚至表征了哈密顿-雅可比方程的整个结构。并给出了该结构的线性控制理论解释。

Firstly, the theory of partialdifferential equations of the 1st order is outlined in the context of the Hamilt on-Jacobi equation. According to the theory, to solve the Hamilton-Jacobi equation, all one has to do is solve a system of ordinary differential equations and/orintegrate a completely integrable Pfaffian system.Secondly, attention is paid to a special kind of solution, a stabilizing solution, which plays an important role in control theory. The well-known theory by Arimoto and Potter, that is, a necessary and sufficient condition of the existence of a stabilizing solution, is reviewed as a part of the theory of the Hamilton-Jacobi equation. In control theory, we are interested in obtaining as many solutions as possible. To do this, more information on the structure of the Hamilton-Jacobi equation will be necessary, which is indicated by an example.Finally, the theory of the generating function for symplectic transforms is introduced ; this will play a central role in a later subsection. Given a solution to the Hamilton-Jacobi equation, an auxiliary equation is derived that determines the other solutions using the generating function of symplectic transforms. It will also be seen that the auxiliary equation even characterizes the whole structure of the Hamilton-Jacobi equation. The linear control theoretic explanation of this structure will also be provided.