BOUNDARY FEEDBACK STABILIZATION OF AN UNSTABLE HEAT EQUATION ∗ WEIJIU LIU † SIAM J. CONTROL OPTIM. c  2003 Society for Industrial and Applied Mathematics Vol. 42, No. 3, pp. 1033–1043 Abstract. In this paper we study the problem of boundary feedback stabilization for the unstable heat equation ut (x, t)= uxx(x, t)+ a(x)u(x, t). This equation can be viewed as a model of a heat conducting rod in which not only is the heat being diffused (mathematically due to the diffusive term uxx) but also the destabilizing heat is generating(mathematicallyduetotheterm au with a> 0). Weshowthatfor any givencontinuously differentiable function a and any given positive constant λ we can explicitly construct a boundary feedback control law such that the solution of the equation with the control law converges to zero exponentially at the rate of λ. This is a continuation of the recent work of Boskovic, Krstic, and Liu[IEEE Trans. Automat. Control, 46 (2001), pp. 2022–2028] and Balogh and Krstic [European J. Control, 8 (2002), pp. 165–176]. Key words. heat equation, boundary control, stabilization AMS subject classifications. 35K05, 93D15 DOI. 10.1137/S0363012902402414 1. Introduction. In this paper we continue the study of boundary feedback control of an unstable heat equation u t (x,t)= u xx (x,t)+ μu(x,t) in(0, 1) × (0, ∞). Hereafter, the subscripts denote the derivatives. This equation can be viewed as a model of a heat conducting rod in which not only is the heat being diffused (mathe- matically due to the term u xx ) but also the destabilizing heat is generating (mathe- maticallyduetotheterm μu with μ> 0). Thisfeedbackcontrolproblemwasrecently addressed by Boskovic, Krstic, and Liu in [5], and it was shown that the unstable rod can be exponentially stabilized by a boundary feedback control law if the constant μ< 3π 2 /4; that is, the destabilizing heat generation is not very big. More recently, Balogh and Krstic [3, 4] removed the condition μ< 3π 2 /4 and replaced μ by an arbitrarily large function a(x): u t (x,t)= u xx (x,t)+ a(x)u(x,t) in(0, 1) × (0, ∞). (1.1) They used a backstepping method for the finite difference semidiscretized approxima- tion of the above equation to derive a Dirichlet boundary feedback control law that makes the closed-loop system stable with an arbitrary prescribed stability margin. They showed that the integral kernel in the control law is bounded. However, some problems like the smoothness of the kernel and Neumann boundary control (usually more difficult than the Dirichlet one) were left open. Using a different method, we completely solve these problems by solving a partial differential equation of the kernel * Received by the editors February 8, 2002; accepted for publication (in revised form) February 24, 2003; published electronically July 8, 2003. This work was done while the author was with the University of Cincinnati and was supported by the Taft Memorial Fund. http://www.siam.org/journals/sicon/42-3/40241.html † Department of Mechanical Engineering, Room 3-455C, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139 (weiliu@mit.edu). 1033