Some New Results on Integral-Type Backstepping Method for a Control Problem Governed by a Linear Heat Equation

It is well known in the literature that the design of stabilization controllers for control systems governed by linear heat equations can be achieved by applying the integral-type backstepping transformation. In this paper, its focus is to establish three new results. First, by controllability theory, we show that the choice of kernels is unique in the context of integral-type backstepping transformation. Next, we show that the forward transformation and inverse transformation in the integral-type backstepping method are mutual transformation pair via solving an easily solvable PDE. With this result, the need of finding explicit solutions of kernel equations can be avoided. Finally, by constructing a corresponding LQ problem, we show that the optimal control of the LQ problem is exactly the stabilization control of the heat equation obtained by the integral-type backstepping method.