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We present an optimization-based framework for analysis and control of linearparabolic partial differential equations (PDEs) with spatially varyingcoefficients without discretization or numerical approximation. For controllersynthesis, we consider both full-state feedback and point observation (outputfeedback). The input occurs at the boundary (point actuation). We use positivematrices to parameterize positive Lyapunov functions and polynomials toparameterize controller and observer gains. We use duality and an invertiblestate-variable transformation to convexify the controller synthesis problem.Finally, we combine our synthesis condition with the Luenberger observerframework to express the output feedback controller synthesis problem as a setof LMI/SDP constraints. We perform an extensive set of numerical experiments todemonstrate accuracy of the conditions and to prove necessity of the Lyapunovstructures chosen. We provide numerical and analytical comparisons withalternative approaches to control including Sturm Liouville theory andbackstepping. Finally we use numerical tests to show that the method retainsits accuracy for alternative boundary conditions.

A Convex Sum-of-Squares Approach to Analysis, State Feedback and Output Feedback Control of Parabolic PDEs