Nonlinear Dyn (2010) 59: 681–693 DOI 10.1007/s11071-009-9570-4 ORIGINAL PAPER Feedback stabilization of a class of nonlinear second-order systems Pawel Skruch Received: 25 March 2009 / Accepted: 23 July 2009 / Published online: 15 August 2009 © Springer Science+Business Media B.V. 2009 Abstract The goal of this paper is to study stabiliza- tion techniques for a system described by nonlinear second-order differential equations. The problem is to determine the feedback control as a function of the state variables. It is shown that the following con- trollers can asymptotically stabilize the system: lin- ear position feedback, linear velocity feedback and a group of nonlinear feedbacks. The stability of the cor- responding closed-loop system is proved by imposing a suitable Lyapunov function and then using LaSalle’s invariance principle. The results of numerical com- putations are included to verify theoretical analysis and mathematical formulation. Some application ex- amples from robotics, mechanics and electronics are presented. Keywords Nonlinear dynamical system · Stability · Lyapunov functions · Nonlinear control · Feedback stabilization 1 Introduction The stabilization of nonlinear second-order systems finds applications in mechanics, electronics and robot- P. Skruch ( ) Department of Automatics, AGH University of Science and Technology, Al. Mickiewicza 30/B1, 30-059 Krakow, Poland e-mail: pawel.skruch@agh.edu.pl ics [25]. As examples we can consider robots with flexible links, vibrating structures such as beams, buildings and bridges, electrical circuits, oscillators, synchronous machines, etc. Nonlinear problems are of interest to many scientists because most physical systems are nonlinear in nature. On the other hand, nonlinear equations are difficult to analyze and they give rise to interesting phenomena such as chaos [17]. The goal of this paper is to study stabilization tech- niques for a class of nonlinear systems. Such class of the systems is described by nonlinear second-order differential equations. Based on the recent results re- garding stabilization of linear oscillatory systems [22] there are constructed feedback control laws which as- ymptotically stabilize the zero equilibrium point of the nonlinear system. The stability of second-order systems both in finite and infinite dimensional cases has been already stud- ied in the past. More recently, in [9, 10], the dynam- ics and the stability of LC ladder network by inner re- sistance, by velocity feedback and by first-order dy- namic feedback have been studied. Control problems for undamped second-order systems are discussed in [2, 11, 12]. In [21], the class of nonlinear controllers is proposed to stabilize damped gyroscopic systems. In the papers [13, 23] the stabilization problem of nonlin- ear one-dimensional oscillator is considered. The ar- ticle [19] copes with another control problem of the equivalent second-order dynamical system, similar to the stabilization problems. The article [20] is also de-