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.