  In this paper we use SOS and SDP to design output feedback controllers for a
class of one-dimensional parabolic partial differential equations with point
measurements and point actuation. Our approach is based on the use of SOS to
search for positive quadratic Lyapunov functions, controllers and observers.
These Lyapunov functions, controllers and observers are parameterized by linear
operators which are defined by SOS polynomials. The main result of the paper is
the development of an improved class of observer-based controllers and evidence
which indicates that when the system is controllable and observable, these
methods will find a observer-based controller for sufficiently high polynomial
degree (similar to well-known results from backstepping).
