Suppose I have the following nonlinear coupled dynamic system

\begin{align*} &\dot{x}_1 = f_1(x_1,x_2)\\ &\dot{x}_2 = f_2(x_2) + u \end{align*}where $x_1\in \mathbb{R}^{n_1}$, $x_2\in \mathbb{R}^{n_2}$, and $u\in \mathbb{R}^{n_2}$. Suppose $f(0,0)=0$ and $u$ is the control input.

Suppose I want to study the stability of the above system w.r.t the state $x_1$ at the $x_1=0$. So consider $V(x_1)$ and suppose that

1. $V(x_1)>0$ for all $x_1 \neq 0$ and $V(x_1)=0$ if $x_1=0$.
2. $\dot{V}(x_1)=(\nabla_{x_1}V)^T \dot{x}_1=(\nabla_{x_1}V)^Tf_1$.

Now, I am confused how to proceed.

1. If $\dot{V}(x_1)=(\nabla_{x_1}V)^T \dot{x}_1=(\nabla_{x_1}V)^Tf_1<0$ for all $x_1 \neq 0$, then can we still say that the system is stable w.r.t $x_1$ at $x_1=0$?
2. The point I am confused is that the control input affect $x_2$ directly; however, $u$ affect $x_1$ by $x_2$. So not sure how to use the Lyapunov theory here correctly.

Can anyone advise me this or refer me some related reference/paper?

Thanks!

Suppose I have the following nonlinear coupled dynamic system

\begin{align*} &\dot{x}_1 = f_1(x_1,x_2)\\ &\dot{x}_2 = f_2(x_2) + u \end{align*}where $x_1\in \mathbb{R}^{n_1}$, $x_2\in \mathbb{R}^{n_2}$, and $u\in \mathbb{R}^{n_2}$. Suppose $f(0,0)=0$ and $u$ is the control input.

Suppose I want to study the stability of the above system w.r.t the state $\textbf{x}_1$ only at the $x_1=0$. So consider $V(x_1)$ and suppose that

1. $V(x_1)>0$ for all $x_1 \neq 0$ and $V(x_1)=0$ if $x_1=0$.
2. $\dot{V}(x_1)=(\nabla_{x_1}V)^T \dot{x}_1=(\nabla_{x_1}V)^Tf_1$.

Now, I am confused how to proceed.

1. If $\dot{V}(x_1)=(\nabla_{x_1}V)^T \dot{x}_1=(\nabla_{x_1}V)^Tf_1<0$ for all $x_1 \neq 0$, then can we still say that the system is stable w.r.t $x_1$ at $x_1=0$?
2. The point I am confused is that the control input affect $x_2$ directly; however, $u$ affect $x_1$ by $x_2$. So not sure how to use the Lyapunov theory here correctly.

Can anyone advise me this or refer me some related reference/paper?

Thanks!