Output-Lagrange stability and output asymptotic gain imply OL-IOS

Let the system be forward complete, with state space $\mathcal X^n$, input space $\mathcal U^m$, and output $y(t,x_0,u)$. Let $\mathcal K_\infty$ denote the class of continuous, strictly increasing, unbounded functions vanishing at zero, and let $\mathcal N$ denote the class of continuous, nondecreasing functions vanishing at zero. Assume that there exist $\sigma\in\mathcal K_\infty$ and $\mu\in\mathcal N$ such that, for every $x_0\in\mathcal X^n$ and every $u\in\mathcal U^m$, the corresponding solution satisfies

$$|y(t,x_0,u)|\leq\max\left\{\sigma(|h(x_0)|),\ \sigma(\|u_{[0,t]}\|)\right\},\quad \forall t\geq 0,$$

and

$$\lim_{t\to+\infty}|y(t,x_0,u)|\leq\mu(\|u\|).$$

Output-Lagrange stability and output asymptotic gain imply OL-IOS. Under these assumptions, the system is OL-IOS.

This conjecture asks whether output-Lagrange stability together with the output asymptotic gain property suffices for output-Lagrange input-to-output stability. The question concerns a solutions-based characterization of IOS for forward-complete time-delay systems; its resolution is not supplied in the source.