Asymptotic Behavior Analysis of a Heat Equation Coupled with a Fractional Ordinary Differential System

State feedback controller design of fractional ordinary differential equations coupled with a fractional reaction–advection–diffusion equation with delay

“…As far as we know, there are a few results about stabilization of fractional ODE-PDE systems. Most of those results focused on dealing with fractional coupled systems without delay where the PDE subsystems were reaction–diffusion systems (Amiri et al, 2021; Chen et al, 2020) or a Riemann–Liouville fractional coupled system without delay (Hou et al, 2022). However, in practical dynamic systems, time delays are inevitable and always affect dynamical behaviors of systems.…”

“…In addition, a small delay may be enough to destabilize some systems (Hale, 1997). Thus, we consider fractional coupled systems (2.1)–(2.5) in which space-dependent coefficients, a convective term, and a delay are appended in the fractional PDE subsystem, which can be viewed as a supplement and extension of the systems (Amiri et al, 2021; Chen et al, 2020; Hou et al, 2022).…”

“…Fractional ODE-PDE systems have a rich physical background, such as they can represent the motion of a ship in a continuous medium (fractional fluid). There are a few results of fractional ODE-PDE systems (Amiri et al, 2021; Chen et al, 2020; Hou et al, 2022). Most of those results focused on dealing with fractional coupled systems without delay where the PDE subsystems were reaction–diffusion systems or a Riemann–Liouville fractional coupled system.…”

“…Most of those results focused on dealing with fractional coupled systems without delay where the PDE subsystems were reaction–diffusion systems or a Riemann–Liouville fractional coupled system. For example, Hou et al (2022) investigated asymptotic behavior of a heat equation coupled with a fractional ordinary differential system in Riemann–Liouville sense by integer-order Lyapunov function method, which cannot be used to analyze Caputo’s sense systems. By backstepping transformation and fractional Lyapunov method, Chen et al (2020) studied the stability of the following system…”