Boundary Feedback Stabilization of an Unstable Heat Equation

In this paper we study the problem of boundary feedback stabilization for the unstable heat equation u t (x,t) = u xx (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 u xx ) but also the destabilizing heat is generating (mathematically due to the term a u with a >0). We show that for any given continuously differentiable function a and any given positive constant 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 . 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].