Prescribed-time boundary control of second-order hyperbolic PDEs modeled flexible string systems via backstepping design

This paper presents a boundary control scheme for prescribed-time (PT) stable flexible string systems via the backstepping method, and the dynamics of such systems modeled by Hamilton’s principle is described as second-order hyperbolic partial differential equations (PDEs). Initially, to construct a boundary controller with PT stabilization capacity, a PT-stable hyperbolic PDEs system with time-varying coefficient is chosen as the target system, and a corresponding Volterra integral transform with a time-varying kernel function is considered. Then, to identify the boundary controller, the well-posedness of the kernel equation is derived by means of successive approximation and mathematical induction, and the upper bound of the kernel function is estimated. Furthermore, the invertibility of the transform is proved with the help of a similar process for the kernel function. Subsequently, the PT stability of the closed-loop system is proved by the PT stability of the target system and the reversible integral transform. Finally, the simulation results demonstrate the effectiveness of our scheme.