There is a state equilibrium point $X_e$ that satisfies $f(x_e)=0$,
Non-linear system,a state equation:$\dot X=f(X)$,where f(X) has a continuous partial derivative for X,has a equilibrium point $X_e$,and then:$\dot X=f(x_e)+\frac{\partial \vec{f}}{\partial \vec{X}^T}_{(X_e)}(X-X_e)+R_{(X-X_e)}=A(X-X_e)+R_{(X-X_e)}$,where Matrix A is a Jacobian matrix:$A=\begin{bmatrix}\frac{\partial f_1}{\partial X_1}&\cdots&\frac{\partial f_1}{\partial X_n}\\ \vdots&\ddots&\vdots\\ \frac{\partial f_n}{\partial X_1}&\cdots&\frac{\partial f_n}{\partial X_n}\end{bmatrix}_{(X_e)}$
Based on the energy analysis, a positive definite energy functional is introduced to describe the system and observe whether the functional is gradually decaying with time to determine the stability of the system.
Attention! Lyapunov's second method is a sufficient condition for determining stability. This means that if such a functional is not found, the system may also be stable.
System state equation:$\dot X=AX$ is necessarily and sufficiently asymptotic stability at the equilibrium point $X_e$:
Hermitian matrix $\forall$Q,there is always a Hermitian matrix P,satisfying the Lyapunov equation:$$PA+A^TP=-Q$$. If the matrix P is positive, the system state is stable and the system positive definite energy functional is $V(x)=X^TPX$.
Matlab code:
P=lyap(A,Q)
A=[1 2;1 4];
PropeValue=eig(A)
P=lyap(A,eye(2));
detP=det(P)
PropeValue = 0.4384 4.5616 detP = 0.1300
System state equation:$\dot X(t)=A(t)X(t)$ is necessarily and sufficiently asymptotic stability at the equilibrium point $X_e$:
Hermitian matrix $\forall$Q,there is a Hermitian matrix P,satisfying the Lyapunov equation:$$\begin{cases}P(t)A(t)+A^T(t)P(t)+\dot P(t)=-Q(t)& t\leq t_f \\ P(t_f)>0\end{cases}$$. If the matrix P is positive, the system state is stable and the system positive definite energy functional is $V(x,t)=X^TP(t)X$.
Known by the theory of Ricatti matrix differential equations:$$P(t)=\Phi^T(t_f,t)P(t_f)\Phi(t_f,t)+\int_t^{t_f}\Phi^T(\tau,t)Q(\tau)\Phi(\tau,t)d\tau$$
System state equation:$X(k+1)=GX(k)$ is necessarily and sufficiently asymptotic stability at the equilibrium point :
Hermitian matrix $\forall$Q,there is a Hermitian matrix P,satisfying the Lyapunov equation:$$G^TPG-P=-Q$$. If the matrix P is positive, the system state is stable and the system positive definite energy functional is $V(x,t)=X^TP(t)X$。
Matlab code:
P=dlyap(G,Q)
In a linear system, if the state point $X_e$ is the equilibrium point, $X_e$ is the system's only equilibrium point. However, in a nonlinear system, the system may have multiple local asymptotically stable equilibrium points (attractors), and there are also multiple unstable equilibrium points (isolators). Lyapunov's second method provides sufficient sufficiency, but it is not necessary.
Nonlinear time-invariant continuous system equation of state:$\dot X(t)=f(X)$,a equilibrium point $X_e$,where f(X) has a continuous partial derivative for X,is necessarily and sufficiently asymptotic stability: $$-Q=J+J^T<0$$ where,$J=\frac{\partial f}{\partial X^T}$ is a Jaccobian matrix,and the positive definite energy functional of the system is $V(x)=\dot X^T\dot X=f^T(X)f(X)$
Nonlinear system state equation:$\dot X=AX+Bf(X)$,a equilibrium point $X_e=0$. if it satisfies $K_{i1}X\leq f_i(X)\leq K_{i2}X$,the equilibrium point $X_e=0$ is the local asymptotic stability point of the system。