The stability of a wave equation with delay-dependent position

Abstract

In this paper, we consider a 1D wave equation with time delay in the information transmission in which the delay depends on position. We mainly discuss the solvability and stability of this time-delay system. Using the analytic method, we prove that if the largest time delay equals to 1, the system equation is insolvable in the sense of exponential bounded solution. If the largest time delay is not equal to 1, we prove that the system is always solvable by the Riesz basis approach. Further, we discuss the stability of this system when it is solvable. We show that if the delay time is smaller than 1 and the feedback gain constant also is small enough, the system is stable exponentially, and when delay is large or the gain constant is unsuitable, the system is unstable. In particular, we obtain a sufficient condition for the exponential stability of the system. Finally, we give some simulations for the stable region.