Consider the following nonlinear system: \begin{align} \dot{x} &= -axy \\ \dot{y} &= axy - by \end{align} It's easy to check that this system has a continuum of equilibria (of the form $(x,0)$). I'm trying to prove the perturbed version of this system \begin{align} \dot{x} &= -axy +u_1(t)\\ \dot{y} &= axy - by+u_2(t) \end{align} is input-to-state stable, i.e, there exist $\beta \in KL$ and $\gamma \in K$ such that
$$ |x(t)|\leq \beta (|x_{0}|,t)+\gamma (\|u\|_{\infty }). $$
I'm not sure how to do that since the system doesn't have an isolated equilibria and most of the results for input-to-state stability seem to be concerned for that particular case. Moreover, I'm not sure how to find a Lyapunov function for the system.
I would suggest to look at the literature on ecological and epidemiological systems as many of those systems have this structure. In fact, this model is the model of a reaction network with mass-action kinetics. So, you may also look at the literature on that topic.
Also, if you are actually working on that, you may need to explicitly consider the positivity of the variables. For instance, you may take the Lyapunov function $V(x,y)=x+y$. In this case, you have $\dot{V}(x,y)=-by$ and you can prove the asymptotic stability of the continuum of equilibrium points. Perhaps, you can use a variation of this Lyapunov function for your purpose.