隐函数存在的充分必要条件

采用状态方程的级数解形式分析系统能控性，它的关键是隐函数存在的判别问题。但常用隐函数存在性定理不满足能控性分析的需要。从常用隐函数存在性定理出发，放宽对隐函数唯一、连续、连续可微等性质的限制，利用解析函数的性质将其展拓，得到广义隐函数存在的充分必要条件和广义隐函数的分布特征。它是定常解析非线性系统弱能控充要条件的证明和受控对称性分析的基础，并在定常非线性系统的能控性分析中得到应用。

Sufficient and necessary conditions of existence of implicit functions

If the general series solutions of the state equations are used as the essential basis to analyze the controllable property of a constant parameter, analytic and nonlinear system, the main problem is the existence of implicit function. These series solutions have the form of implicit functions of time, states, controls and their differentials along the time. So the analysis of controllable property is to find the solutions of the implicit functions. But the implicit function theorem can't satisfy the requirement of the controllability. This article uses the implicit function theorem and the properties of analytic functions, with less limitations for the uniqueness, continuity, continuous differentiability of the implicit functions, and gets the sufficient and necessary conditions of the existence of implicit functions by simplifying them. These conditions can be used to prove the sufficient and necessary conditions of the weak controllable property. In addition, the results of the analysis of the general implicit function can also be used in the analysis of the symmetrical property of general constant parameter, analytic and nonlinear systems. And it can be applied in the controlbility analysis of constant nonlinear system.