Stabilization of a coupled second order ODE-wave system

Abstract

This paper considers the stabilization of a coupled second order ODE-wave system, where the ODE dynamics contain the solution of the wave equation at an intermediate point. We design a stabilizing feedback controller by choosing a suitable target system and backstepping transformation. The backstepping transformation is defined in terms of several kernel functions, for which we establish existence, uniqueness and smoothness properties. We also prove exponential stability for the resulting closed-loop system. Finally, the effectiveness of the proposed feedback controller is verified via a numerical example.

Infinite‐dimensional extended state observer based stabilization for an ODE‐wave cascaded system with control matched disturbance

“…Recent years, the control design for ODE-PDE cascaded system receives more and more attentions [6][7][8]. Backstepping method, developed by Krstic, is a powerful tool for the design of exponential stability control law for ODE-PDE cascaded systems, such as ODE-Schrödinger equations by Ren, Wang and Krstic [9], Tang and Xie's ODE-heat equations [10,11], and ODE-wave equations [12][13][14].…”

Dynamic stabilization for a cascaded beam PDE–ODE system with boundary disturbance

“…For PDE-ODE cascaded system without considering the boundary disturbance, backstepping approach has been successfully used to the stabilization of heat-ODE, 1,2,6,7 Schrödinger-ODE, 8,9 and wave-ODE. 10,11 However, on one hand,the actual expressions of system models are rather difficult (or even impossible) to obtain in practice engineering; on the other hand, disturbance caused by complex external circumstance is also unavoidable. Accordingly, uncertain disturbance should be taken into consideration.…”

“…Since the PDEs and ODEs coexist, the control strategies for single PDE or ODE system may lose efficacy for the coupled PDE–ODE one. For PDE–ODE cascaded system without considering the boundary disturbance, backstepping approach has been successfully used to the stabilization of heat‐ODE, 1,2,6,7 Schrödinger‐ODE, 8,9 and wave‐ODE 10,11 . However, on one hand,the actual expressions of system models are rather difficult (or even impossible) to obtain in practice engineering; on the other hand, disturbance caused by complex external circumstance is also unavoidable.…”