Adaptive boundary control for wave PDE on a domain with moving boundary and with unknown system parameters in the boundary dynamics

In this paper, we apply the technique of non-constant delay control to a problem of velocity stabilization for a system whose domain is time-varying. This system is modeled as a wave PDE on a domain with a moving boundary. The problem is transformed first into an LTI form describing the boundary dynamics. A time-varying input delay and unknown parameters characterize this uncertain linear system. From there, we use the backstepping design via a transport PDE to construct an adaptive boundary controller. The boundary states are assumed to be the only measurements available for feedback. The stability analysis is given for the closed-loop system, and the boundedness of the state is proven. The paper concludes with a numerical simulation illustrating and contrasting the performance of the derived controller.