Journal of the Franklin Institute 348 (2011) 2142–2155 Stabilization for a coupled PDE–ODE control system $ Shuxia Tang, Chengkang Xie à School of Mathematics and System Science, Southwest University, Chongqing 400715, China Received 12 May 2010; received in revised form 13 January 2011; accepted 7 June 2011 Available online 1 July 2011 Abstract A control system of an ODE and a diffusion PDE is discussed in this paper. The novelty lies in that the system is coupled. The method of PDE backstepping as well as some special skills is resorted in stabilizing the coupled PDE–ODE control system, which is transformed into an exponentially stable PDE–ODE cascade with an invertible integral transformation. And a state feedback boundary controller is designed. Moreover, an exponentially convergent observer for anti-collocated setup is proposed, and the output feedback boundary control problem is solved. For both the state and output feedback boundary controllers, exponential stability analyses in the sense of the corresponding norms for the resulting closed-loop systems are given through rigid proofs. & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. 1. Introduction In control engineering, systems modeled by ordinary differential equations (ODE) are common. Over the past decades of years, systems modeled by partial differential equations (PDE) have been popular too. Recently, coupled systems have been active areas of research. Examples can be found in control problems of electromagnetic coupling, mechanical coupling and chemical reaction coupling. Some results on controllability of coupled PDE–PDE systems have been established (see, e.g., [12–14] ). However, the problem of feasible controllers and www.elsevier.com/locate/jfranklin 0016-0032/$32.00 & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2011.06.008 $ This work is supported by the Fundamental Research Funds for the Central Universities under contract XDJK2009C099. à Corresponding author. E-mail addresses: tkongzhi@swu.edu.cn (S. Tang), cxie@swu.edu.cn (C. Xie).