  Recently finite-dimensional observer-based controllers were introduced for
the 1D heat equation, where at least one of the observation or control
operators is bounded. In this paper, for the first time, we manage with such
controllers for the 1D heat equation with both operators being unbounded. We
consider Dirichlet actuation and point measurement and use a modal
decomposition approach via dynamic extension. We suggest a direct Lyapunov
approach to the full-order closed-loop system, where the finite-dimensional
state is coupled with the infinite-dimensional tail of the state Fourier
expansion, and provide LMIs for finding the controller dimension and the
resulting exponential decay rate. We further study sampled-data implementation
of the controller under sampled-data measurement. We use Wirtinger-based,
discontinuous in time, Lyapunov functionals which compensate sampling in the
finite-dimensional state. To compensate sampling in the infinite-dimensional
tail, we use a novel form of Halanay's inequality, which is appropriate for
Lyapunov functions with jump discontinuities that do not grow in the jumps.
Numerical examples demonstrate the efficiency of the method.
