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

TL;DR

This paper develops a boundary control method using backstepping to achieve prescribed-time stabilization of flexible string systems modeled by second-order hyperbolic PDEs, with theoretical proofs and simulations confirming effectiveness.

Contribution

It introduces a novel backstepping boundary control scheme for prescribed-time stabilization of hyperbolic PDEs with time-varying coefficients, including kernel equation solutions and stability proofs.

Findings

Successfully stabilizes flexible string systems within prescribed time

Provides a well-posed kernel equation with bounded kernel functions

Simulation results confirm the effectiveness of the control scheme

Abstract

This paper presents a boundary control scheme for prescribed-time (PT) stable of flexible string systems via 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 time-varying kernel function is considered. Then, to identify the boundary controller, the well-posedness of kernel equation is derived by means of successive approximation and mathematical induction, and the upper bound of kernel function is estimated. Furthermore, the inverse transform is proved with the help of a similar process for kernel function. Subsequently, the PT stability of closed-loop system is proved by PT stability of target system and reversible integral transform. Finally, the simulation results demonstrate the effectiveness of our scheme.