- 非线性系统的稳定性可能只有局部性质
- 且李雅普诺夫第二法只能给出非线性系统渐近稳定的充分条件，而非必要条件，求解较为简单，如果无法找到李雅普诺夫函数才使用下述方法。
一、雅可比矩阵法/克拉索夫斯基法
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。
\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})\quad \mathbf{J}(\mathbf{x})= \dfrac{\partial \mathbf{f}(\mathbf{x})}{\partial \mathbf{x}} = \begin{pmatrix} \dfrac{\partial f_1}{\partial x_1} & \dfrac{\partial f_1}{\partial x_2} & \cdots & \dfrac{\partial f_1}{\partial x_n} \\ \dfrac{\partial f_2}{\partial x_1} & \dfrac{\partial f_2}{\partial x_2} & \cdots & \dfrac{\partial f_2}{\partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \dfrac{\partial f_m}{\partial x_1} & \dfrac{\partial f_m}{\partial x_2} & \cdots & \dfrac{\partial f_m}{\partial x_n} \end{pmatrix} \end{align}$$ 在**原点渐近稳定**的 **充分条件**是：对于任意给定的实对称矩阵 $P$，使得矩阵 $Q(x)=-[J^{T}(\mathbf{x})P+PJ(\mathbf{x})]$ 为正定矩阵，李雅普诺夫函数： $V(\mathbf{x})=\mathbf{f}^{T}(\mathbf{x})P\mathbf{f}(\mathbf{x})$ 实际计算一般设 $P=I$，$Q>0$ 则系统在平衡点处渐近稳定 $$\begin{align} Q(\mathbf{x})=-\left[J^{T}(\mathbf{x})+J(\mathbf{x})\right]>0 \end{align}$$ $$\begin{align} V(\mathbf{x})=\mathbf{f}^{T}(\mathbf{x})\mathbf{f}(\mathbf{x})\to \infty \quad (\left\lvert \left\lvert \mathbf{x} \right\rvert\right\rvert \to \infty) \end{align}$$ 则进一步有大范围渐近稳定 ### ~~二、变量梯度法/舒茨-基布逊法~~ 理论基础：[[梯度\|梯度]] [[旋度\|旋度]] $$\begin{align} \nabla V = \dfrac{\partial V}{\partial \mathbf{x}} =\begin{pmatrix} \dfrac{\partial V}{\partial x_{1}}\\ \dfrac{\partial V}{\partial x_{2}} \\ \vdots\\\dfrac{\partial V}{\partial x_{n}} \end{pmatrix} =grad \; V(\mathbf{x}) \end{align}$$ $$\begin{align} \dot{V}(\mathbf{x})= (\dfrac{\partial V}{\partial x_{1}}, \dfrac{\partial V}{\partial x_{2}},\cdots,\dfrac{\partial V}{\partial x_{n}} ) \begin{pmatrix} \dot{x}_{1} \\\dot{x}_{2}\\ \vdots \\ \dot{x}_{n} \end{pmatrix} =\left[\;\nabla V\;\right]^{T} \mathbf{\dot{x}} \end{align}$$ $$\begin{align} V(\mathbf{x})=\int _{0}^{\mathbf{x}} (\nabla V)^{T}\, d\mathbf{\mathbf{x}}=\int _{0}^{x_{1}(x_{2}=\cdots=x_{n}=0)} \nabla V_{1}\, dx_{1} +\int _{0}^{x_{2}(x_{1}=x_{1},x_{3}=\cdots=x_{n}=0)} \nabla V_{2}\, dx_{2} +\cdots+ \int _{0}^{x_{n}(x_{1}=x_{1},\cdots,x_{n-1}=x_{n-1})} \nabla V_{n}\, dx_{n} \end{align}$$