  Recently, a constructive method was suggested for finite-dimensional
observer-based control of 1D linear heat equation, which is robust to
input/output delays. In this paper, we aim to extend this method to the 2D case
with general time-varying input/output delays (known output delay and unknown
input delay) or sawtooth delays (that correspond to network-based control). We
use the modal decomposition approach and consider boundary or non-local sensing
together with non-local actuation, or Neumann actuation with non-local sensing.
To compensate the output delay that appears in the infinite-dimensional part of
the closed-loop system, for the first time for delayed PDEs we suggest a vector
Lyapunov functional combined with the recently introduced vector Halanay
inequality. We provide linear matrix inequality (LMI) conditions for finding
the observer dimension and upper bounds on delays that preserve the exponential
stability. We prove that the LMIs are always feasible for large enough observer
dimension and small enough upper bounds on delays. A numerical example
demonstrates the efficiency of our method and shows that the employment of
vector Halanay's inequality allows for larger delays than the classical scalar
Halanay inequality for comparatively large observer dimension.
