Delayed finite-dimensional observer-based control of 1-D parabolic PDEs

In our recent paper a constructive method for finite-dimensional observer-based control of 1-D linear heat equation was suggested. In the present paper we aim to extend this method to the case of input/output general time-varying delays or sawtooth delays (that correspond to network-based control). We assume known measurement delays and, for the first time under observer-based control of PDEs, unknown input delays. We use a modal decomposition approach, and consider boundary or non-local sensing together with non-local actuation, or Dirichlet actuation with non-local sensing. The dimension of the controller is equal to the number of unstable modes, whereas the observer may have a larger dimension N. Under the Dirichlet actuation we present two methods: a direct one that manages with time-varying input and output delays, and a dynamic-extension-based one that treats constant input and time-varying output delays. To compensate the fast-varying output delay (without any constraints on the delay derivative) that appears in the infinite-dimensional part of the closed-loop system, we combine Lyapunov functionals with Halanay's inequality. For the slowly-varying output delay (with the delay derivative smaller than d<1), we suggest a direct Lyapunov method. We provide LMIs for finding N and upper bounds on the delays that preserve the exponential stability.