  In this paper we use optimization-based methods to design output-feedback
controllers for a class of one-dimensional parabolic partial differential
equations. The output may be distributed or point-measurements. The input may
be distributed or boundary actuation. We use Lyapunov operators, duality, and
the Luenberger observer framework to reformulate the synthesis problem as a
convex optimization problem expressed as a set of Linear-Operator-Inequalities
(LOIs). We then show how feasibility of these LOIs may be tested using
Semidefinite Programming (SDP) and the Sum-of-Squares methodology.
