Boundary stabilization of quasi-linear hyperbolic systems with varying time delay - Gugat, Martin (SIAM Journal on Control and Optimization, 2025)
In this paper we consider the boundary feedback stabilization of a quasi-linear hy- perbolic system of balance laws. At one end of the space interval, there is a reflecting boundary condition. At the other end a stabilizing feedback law with a varying time-delay is prescribed. We present sufficient conditions for the exponential stability of the system. We show that exponential stabilization is possible if the product of the length of the interval and an upper bound for the source term is sufficiently small. We also show that if the product of the length of the interval and a lower bound for the source term is sufficiently large, the system is unstable. Our analysis is based on Lya- punov functions with weights that are given by hyperbolic functions that generalize the well-known exponential weights. Compared with previous contributions, we obtain conditions that can be veri- fied more easily in terms of the system parameters. Our results show that for sufficiently short space intervals, and also with varying time-delay, exponential stabilization is possible with appropriately chosen feedback gains that depend on the maximal value of the time-delay and the maximal absolute value of its derivativ
Keywords: boundary stabilization; feedback law; varying delay; quasi-linear hyperbolic systems; boundary controL