Regional stabilization of a fractional output for a class of time-delayed distributed bilinear systems

Abstract

In this research, we examine the regional stabilization problem of the Riemann–Liouville spatial fractional output of order $\alpha \in [0,\ 1],$ for a class of time-delayed distributed bilinear systems evolving on the domain $\Omega .$ The goal is to find a control that depends on $\alpha$ and the subregion $\omega \subset \Omega$ allowing the stabilization of the considered system and generalizing the obtained results ( see Tsouli et al. (Int J Control 94(8):2065–2071, 2021), El Houch et al. (Int J Control 94(6):1693–1703, 2021), Hamidi et al. (J Dyn Control Syst 26(2):243–254, 2020), Zerrik et al. (IFAC-PapersOnLine 55(12):729–734, 2022)). In particular, if $\omega = \Omega$ and $\alpha = 0,$ we obtain the global stabilization over the evolution domain. On the other hand, we get the gradient regional stabilization of the output system with $\alpha = 1$ for every subregion $\omega$ of $\Omega .$ Hence, we display that the obtained control allows us to generalize the last cases and stabilizes both weakly and strongly the fractional output for each values of $\alpha \in [0,\ 1]$ under some sufficient conditions. Finally, we provide computational simulations to check the capability of the acquired stabilization results.