Unstable Reaction-Diffusion PDE with Input Delay

Abstract

In this chapter and in Chapter 20 we introduce the problems of stabilization of PDEPDE cascades. First, we deal with PDEs with input delays (the reaction-diffusion PDE in this chapter, and the antistable wave PDE in Chapter 20). Then we deal with cascades of unstable heat and wave PDEs (in either order) in Chapter 20.

Stability analysis for cascades of stable PDEs from different classes, when interconnected through a boundary, virtually explodes in complexity despite the seemingly simple structure where one PDE is autonomous and exponentially stable and feeds into the other PDE. The difficulty arises for two reasons. One is that the connectivity through the boundary gives rise to an unbounded input operator in the interconnection. The second reason is that the two subsystems are from different PDE classes, with different numbers of derivatives in space or time (or both). This requires delicate combinations of norms in the Lyapunov functions for the overall systems.