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

Output Feedback Control of Inhomogeneous Parabolic PDEs with Point Actuation and Point Measurement using SOS and Semi-Separable Kernels