Boundary Feedback Control of Unstable Heat Equation with Space and Time Dependent Coefficient

Abstract

In this paper, we study the control of the linear heat equation with a space and time dependent coefficient function by the Dirichlet and Neumann boundary control laws. This equation models the heat diffusion and space, time dependent heat generation in a one dimensional rod. Without control, the system is unstable if the coefficient function is positive and large. With boundary control based on the state feedback, we show that for the time analytic coefficient $a(x,t)$, the exponential stability of the system at any rate can be achieved. It is further shown that both the control of the Dirichlet and Neumann boundary value systems can be stabilized using this method. In doing this, the control kernels are {\it explicitly} calculated as series of approximation and they are used in the simulations. The numerical simulation confirmed the theoretical arguments and the controllability of the system. The possible future works are discussed at the end. This is the continuation of the recent work of Liu [{\it SIAM J. Cont. Optim.} vol. 42, pp. 1033-1043] and Smyshlyaev and Krstic [accepted by {\it Automatica}].