How do I investigate input-to-state stability for $\dot{x}=-(1+u)x^3$

Question

I am wondering how to solve this problem. I am supposed to investigate the ISS stability. But I am having trouble seeing where to start.

How do I investigate input-to-state stability for $\dot{x}=-(1+u)x^3$

I have that:

for when $u(t)$ is identical to 0, the origin is asympotically stable and that for a bounded input $u(t)$, every solution $x(t)$ is bounded.

Can anyone give me a hint or clue on where to start?

Answer 1

The system is not input to state stable. Take the constant input $u = -2$:

$$ \dot{x} = -(1 + u) x^3 = -x^3 + 2 x^3 = x^3 $$

So although the input is bounded for all time the state can grow unbounded.