Boundary feedback stabilization of a class of coupled hyperbolic equations with nonlocal terms

This paper solves the problem of boundary feedback stabilization of a class of coupled ordinary differential equations-hyperbolic equations with boundary, trace, and integral nonlocal terms. Using the backstepping approach, the controller is designed by formulating an integral operator, whose kernel is required to satisfy a coupled hyperbolic partial integral differential equation. By applying the method of successive approximations, the kernel's well-posedness is given. We prove the exponential stability of the origin of the system in a suitable Hilbert space. Moreover, a wave system with nonlocal terms is stabilized by applying the above result.