Conjecture on equivalence of average-dwell-time stability notions

Let $\tau>0$ and consider the switched system defined by System~

. For $N_0\in\mathbb{N}$, let $\mathcal{S}_{\mathrm{adw}}(\tau,N_0)$ denote the class of switching signals satisfying the average dwell-time condition with parameters $\tau$ and $N_0$. A system is **uniformly globally bounded** ($\tau\text{-UGB}$) if its trajectories are uniformly globally bounded, and it is **uniformly globally asymptotically stable** (UGAS) or **uniformly globally exponentially stable with rate $\rho$** ($\text{UGES}_\rho$) with respect to a switching-signal class in the usual sense. Given $\rho>0$, a system is $\tau\text{-UGEB}_\rho$ when it has the corresponding uniform global exponential boundedness property with rate $\rho$. **Average-dwell-time stability equivalence conjecture.** For every $\tau>0$, System~

is $\tau\text{-UGB}$ if and only if it is UGAS with respect to $\mathcal{S}_{\mathrm{adw}}(\tau,N_0)$ for any $N_0\in\mathbb{N}$. Given $\rho>0$, it is $\tau\text{-UGEB}_\rho$ if and only if it is $\text{UGES}_\rho$ with respect to $\mathcal{S}_{\mathrm{adw}}(\tau,N_0)$ for all $N_0\in\mathbb{N}$.

This conjecture proposes a converse Lyapunov-type equivalence between stability properties of a switched system and stability uniformly over average-dwell-time switching classes. The paper presents partial analysis but does not establish a complete answer, so the equivalence remains open.