Lyapunov Adaptive Stabilization of Parabolic PDEs— Part II: Output Feedback and Other Benchmark Problems

We deal with parametric uncertainties in boundary conditions or reaction terms involving boundary values. We show how adaptive boundary control problems can be solved using output feedback, for unstable PDEs with infinite relative degree. Boundary sensing is employed, along with a Kreisselmeier type adaptive observer. We also design adaptive boundary controllers for a reaction-advection-diffusion system with all three of the coefficients unknown. Our Lyapunov approach yields parameter estimators that do not require the measurement of any of the spatial derivatives of the controlled variable, which are needed in other approaches. The designs in this paper illustrate the requirement in the Lyapunov approach that parameter projection be used in the update laws. Projection is not used as a robustification tool but to prevent adaptation transients that would require overly conservative restrictions on the size of the adaptation gain.