Question about Lyapunov stability for a nonlinear system with inputs

Question

I am studying Lyapunov stability for systems of the type:

$\dot{x}=f(x)$

and I think I have well understood the theory.

But, suppose now I have a system with an input, so of the type:

$\dot{x}=f(x)+g(x)u$

Can I still use the Lyapunov analysis for analizing the stability of this system?

Answer 1

No, but there are extensions that allow you to do so.

You can look up input to state stability for example. But you will need some assumptions for $u$, for example that the input is bounded.

For example, the system

$$ \dot{x} = -x^3 + xu $$

is input to state stable if $|u| < C$ for any finite positive $C$ because as $x$ gets large the stabilizing $-x^3$ "beats" the $xu$.

But the system

$$ \dot{x} = -x^3 + x^3u $$

is not input to state stable for any bounded input, because with the input $u = 2$ you get

$$ \dot{x} = x^3 $$

which is unstable.

This is of course not very rigorous but there exists a rigorous theory and a lot of literature for this topic. However it gets quickly much more complicated if you have more complicated systems.