Determine if system is input-state stable

Question

I was trying to determine whether the following system is input-state stable (ISS): \begin{align} \dot{x_1} &= -x_1 + (1 - e^{x_2}) \\ \dot{x_2} &= -x_2 - u \end{align}

I tried using the following Lyapunov function but with no luck: \begin{equation} V(x) = \frac{1}{2}x_1^2 + \frac{1}{2}x_2^2 \end{equation}

So I basically have to identify $\alpha_1(||x||)$, $\alpha_2(||x||)$, $W_3(x)$, $\rho(||x||)$

Any suggestions on how I could do it?

DEFINITION: This is the condition that must be fulfilled for a system to be ISS

Answer 1

Your question is missing an assumption on $u$ so I assume it is bounded (otherwise checking ISS doesn't make sense).

Note that $\dot{x}_2$ does not depend on $x_1$, so the system $\dot{x}_2=-x_2 - u$ is ISS (stable LTI systems are always ISS). This means $x_2$ is bounded for all $t$. Therefore $1 - e^{x_2}$ is also bounded for all $t$. So we can consider

$$ \dot{x}_1=-x_1+w $$

where $w$ is a bounded input. This is again a stable LTI system with bounded input so it is ISS. Since this is more general, it also covers the "special" input $w=1-e^{x_2}$ (which is bounded as discussed before).

Therefore the whole system is ISS.