Delay-adaptive control of first-order hyperbolic partial integro-differential equations

We develop a delay-adaptive controller for a class of first-order hyperbolic partial integro-differential equations with an unknown input delay. By employing a transport PDE to represent delayed actuator states, the system is transformed into a transport partial differential equation with unknown propagation speed cascaded with a PIDE. A parameter update law is designed using a Lyapunov argument and the infinite-dimensional backstepping technique to establish global stability results. Subsequently, the effectiveness of the proposed approach was validated through numerical simulations.