Boundary Feedback Stabilization of an Unstable Heat Equation

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 (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 $\l$ 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 $\l$. 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].

[1]  J. A. Burns,et al.  Regularity of feedback opertors for boundary control of thermal processes , 1994 .

[2]  Miroslav Krstic,et al.  Infinite Dimensional Backstepping-Style Feedback Transformations for a Heat Equation with an Arbitrary Level of Instability , 2002, Eur. J. Control.

[3]  Miroslav Krstic,et al.  Boundary control of an unstable heat equation via measurement of domain-averaged temperature , 2001, IEEE Trans. Autom. Control..

[4]  Miroslav Krstic,et al.  Infinite-step backstepping for a heat equation-like PDE with arbitrarily many unstable eigenvalues , 2001, Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148).

[5]  W. A. Day.  A decreasing property of solutions of parabolic equations with applications to thermoelasticity , 1983 .

[6]  R. Triggiani,et al.  The regulator problem for parabolic equations with dirichlet boundary control , 1987 .

[7]  Irena Lasiecka,et al.  Stabilization and Structural Assignment of Dirichlet Boundary Feedback Parabolic Equations , 1983 .

[8]  R. Triggiani,et al.  Stabilization of Neumann boundary feedback of parabolic equations: The case of trace in the feedback loop , 1983 .

[9]  D. Rubio,et al.  A distributed parameter control approach to sensor location for optimal feedback control of thermal processes , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[10]  O. A. Ladyzhenskai︠a︡,et al.  Linear and Quasi-linear Equations of Parabolic Type , 1995 .

[11]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[12]  Roberto Triggiani,et al.  Boundary feedback stabilizability of parabolic equations , 1980 .