Signals, and Systems Backstepping in Infinite Dimension for a Class of Parabolic Distributed Parameter Systems*

Abstract. In this paper a family of stabilizing boundary feedback control laws for a class of linear parabolic PDEs motivated by engineering applications is pre-sented. The design procedure presented here can handle systems with an arbitrary finite number of open-loop unstable eigenvalues and is not restricted to a particu-lar type of boundary actuation. Stabilization is achieved through the design of coordinate transformations that have the form of recursive relationships. The fundamental di‰culty of such transformations is that the recursion has an infinite number of iterations. The problem of feedback gains growing unbounded as the grid becomes infinitely fine is resolved by a proper choice of the target system to which the original system is transformed. We show how to design coordinate transformations such that they are su‰ciently regular (not continuous but Ly). We then establish closed-loop stability, regularity of control, and regularity of solutions of the PDE. The result is accompanied by a simulation study for a lin-earization of a tubular chemical reactor around an unstable steady state