Output-feedback adaptive control for parabolic PDEs with spatially varying coefficients

All of the existing results in adaptive control for parabolic PDEs rely on full state measurement. For the first time, we consider a problem of output feedback stabilization of distributed parameter systems with unknown reaction, advection, and diffusion parameters. Both sensing and actuation are performed at the boundary and the unknown parameters are allowed to be spatially varying. First we construct a special transformation of the original system into the PDE analog of "observer canonical form," with unknown parameters multiplying the measured output. We then use the so-called swapping method for parameter estimation. Input and output filters are implemented so that a dynamic parametrization of the problem is converted into a static parametrization where a gradient estimation algorithm is used. The control gain is computed through the numerical solution of an ordinary integro-differential equation. The results are illustrated by simulations.