Asynchronous-delay boundary control of general hyperbolic PDEs with zero transport speeds

This paper focuses on addressing the challenge of compensating for input delays in a broad class of coupled (n+l+m) systems governed by hyperbolic-type partial differential equations with zero transport velocity. The controller is located at the right boundary of the domain and exhibits different delays. Conventional backstepping design is no longer suitable for asynchronous delay problems. To address the challenges posed by asynchronous delays, we develop a backstepping-based compensator and propose a novel asynchronous-backstepping transformation. The transformation kernels are defined over a square domain and feature a more intricate coupling structure, which presents significant challenges for the analysis of well-posedness. We tackle these problems through the employment of characteristic analysis alongside stepwise approximation techniques. Furthermore, since our stability result is exponential, we further demonstrate that the asynchronous-backstepping transformation guarantees the equivalence of norms between the original system states and the target system states. Finally, a numerical example illustrates the effectiveness of the proposed approach.