李雅普诺夫方法在非线性系统中的应用

理论基础：雅可比矩阵，寻找线性系统李雅普诺夫函数的推广方法。

\begin{array}{r} \dot{x} = f (x) J (x) = \frac{\partial f (x)}{\partial x} = (\begin{array}{c} \frac{\partial f_{1}}{\partial x_{1}} & \frac{\partial f_{1}}{\partial x_{2}} & \dots & \frac{\partial f_{1}}{\partial x_{n}} \\ \frac{\partial f_{2}}{\partial x_{1}} & \frac{\partial f_{2}}{\partial x_{2}} & \dots & \frac{\partial f_{2}}{\partial x_{n}} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ \frac{\partial f_{m}}{\partial x_{1}} & \frac{\partial f_{m}}{\partial x_{2}} & \dots & \frac{\partial f_{m}}{\partial x_{n}} \end{array}) \end{array}

在原点渐近稳定的 充分条件是：对于任意给定的实对称矩阵 $P$ ，使得
矩阵 $Q (x) = - [J^{T} (x) P + P J (x)]$ 为正定矩阵
李雅普诺夫函数： $V (x) = f^{T} (x) P f (x)$

实际计算一般设 $P = I$ ， $Q > 0$ 则系统在平衡点处渐近稳定

\begin{array}{r} Q (x) = - [J^{T} (x) + J (x)] > 0 \end{array}

\begin{array}{r} V (x) = f^{T} (x) f (x) \to \infty (| | x | | \to \infty) \end{array}

则进一步有大范围渐近稳定

理论基础：梯度旋度

\begin{array}{r} \nabla V = \frac{\partial V}{\partial x} = (\begin{array}{c} \frac{\partial V}{\partial x_{1}} \\ \frac{\partial V}{\partial x_{2}} \\ ⋮ \\ \frac{\partial V}{\partial x_{n}} \end{array}) = g r a d V (x) \end{array}

\begin{array}{r} \dot{V} (x) = (\frac{\partial V}{\partial x_{1}}, \frac{\partial V}{\partial x_{2}}, \dots, \frac{\partial V}{\partial x_{n}}) (\begin{array}{c} {\dot{x}}_{1} \\ {\dot{x}}_{2} \\ ⋮ \\ {\dot{x}}_{n} \end{array}) = {[\nabla V]}^{T} \dot{x} \end{array}

\begin{array}{r} V (x) = \int_{0}^{x} (\nabla V)^{T} d x = \int_{0}^{x_{1} (x_{2} = \dots = x_{n} = 0)} \nabla V_{1} d x_{1} + \int_{0}^{x_{2} (x_{1} = x_{1}, x_{3} = \dots = x_{n} = 0)} \nabla V_{2} d x_{2} + \dots + \int_{0}^{x_{n} (x_{1} = x_{1}, \dots, x_{n - 1} = x_{n - 1})} \nabla V_{n} d x_{n} \end{array}