  We present an optimization-based framework for analysis and control of linear
parabolic partial differential equations (PDEs) with spatially varying
coefficients without discretization or numerical approximation. For controller
synthesis, we consider both full-state feedback and point observation (output
feedback). The input occurs at the boundary (point actuation). We use positive
matrices to parameterize positive Lyapunov functions and polynomials to
parameterize controller and observer gains. We use duality and an invertible
state-variable transformation to convexify the controller synthesis problem.
Finally, we combine our synthesis condition with the Luenberger observer
framework to express the output feedback controller synthesis problem as a set
of LMI/SDP constraints. We perform an extensive set of numerical experiments to
demonstrate accuracy of the conditions and to prove necessity of the Lyapunov
structures chosen. We provide numerical and analytical comparisons with
alternative approaches to control including Sturm Liouville theory and
backstepping. Finally we use numerical tests to show that the method retains
its accuracy for alternative boundary conditions.
