Automatica 46 (2010) 1203–1209 Contents lists available at ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica Brief paper Arbitrary decay rate for two connected strings with joint anti-damping by boundary output feedback ✩ Bao-Zhu Guo a,b,c , Feng-Fei Jin a,∗ a Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, PR China b School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa c School of Mathematical Sciences, Shanxi University, Taiyuan 030006, PR China article info Article history: Received 29 September 2009 Received in revised form 21 January 2010 Accepted 23 March 2010 Available online 20 April 2010 Keywords: Wave equation Observer Stability Backstepping Boundary control abstract In this paper, we are concerned with the boundary stabilization of two connected strings with middle joint anti-damping for which all eigenvalues of the (control) free system are located on the right complex plane. We first design an explicit state feedback controller to achieve exponential stability for the closed- loop system. Consequently, we design the output feedback by using infinite-dimensional observer. The backstepping approach is adopted in investigation. It is shown that by using one boundary stabilizer only, the output feedback can make the closed-loop system exponentially stable with arbitrary decay rate. © 2010 Elsevier Ltd. All rights reserved. 1. Introduction The stabilization of one-dimensional connected strings by linear joint feedbacks was initiated in Chen, Coleman, and west (1987). The later studies can be found in Chen and Zhou (1990), Liu (1988), Liu, Huang, and Chen (1989). In Guo and Zhu (1997), the stability of two connected strings under various joint feedbacks was studied by the spectral analysis with the understanding that such a kind of system satisfies the spectrum-determined growth condition, a very special property for the infinite-dimensional systems (Luo, Guo, & Morgul, 1999; Renardy, 1994). More profound result, the Riesz basis property, was first obtained in Xu and Guo (2003) for two connected strings with joint feedbacks. The general Riesz basis property for N-connected strings with joint feedbacks can be found in Guo and Xie (2004) and Guo and Xu (2006). In all these studies aforementioned, the control is in the form of collocated way, that is, the actuator and sensor are located at the same place and have some adjoint property so that the closed- loop system under the proportional output feedback is dissipative. Other studies on the stabilization of connected strings can be found ✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Xiaobo Tan under the direction of Editor Miroslav Krstic. ∗ Corresponding author. Tel.: +86 10 62651443; fax: +86 10 62587343. E-mail addresses: bzguo@iss.ac.cn (B.-Z. Guo), jinfengfei@amss.ac.cn (F.-F. Jin). in Berrahmoune (2004), Khapalov (1997) and Najafi, Sarhangi, and Wang (1997), to name just a few. On the other hand, the non-collocated boundary stabilizing controller for one-dimensional wave equation was designed and the Riesz basis property was analyzed in Guo and Xu (2007). In Krstic, Guo, Balogh, and Smyshlyaev (2008), the non-collocated stabilization was discussed for unstable wave equation. Particular attention should be paid to Krstic (in press) and Smyshlyaev and Krstic (2009) where the anti-stable wave equations with boundary anti-damping were stabilized by using the backstepping method for partial differential equation systems (PDEs). For more examples, we refer to Krstic and Smyshlyaev (2009). In this paper, we shall generalize the result of Smyshlyaev and Krstic (2009) to two connected anti-stable strings with joint anti-damping. The model is described by the following partial differential equation:          w ∗ tt (x, t ) = w ∗ xx (x, t ), x ∈ (0, 1) ∪ (1, 2), t > 0, w ∗ (1 − , t ) = w ∗ (1 + , t ), t ≥ 0, w ∗ x (1 − , t ) − w ∗ x (1 + , t ) = qw ∗ t (1, t ), t ≥ 0, w ∗ (0, t ) = u(t ), t ≥ 0, w ∗ x (2, t ) = 0, t ≥ 0, (1) where w ∗ is the displacement of the string, u is the control input, and q > 0, q  = 2 is the damping constant. This system models two connected strings with joint vertical force anti-damping. Actually, by a result of Guo and Zhu (1997), all eigenvalues of the free system 0005-1098/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2010.03.019