Lyapunov Adaptive Stabilization of Parabolic PDEs - Part I: A Benchmark for Boundary Control

We develop an adaptive controller for a benchmark parabolic PDE controlled from a boundary and containing an unknown destabilizing parameter affecting the interior of the domain. This design departs from prior approaches that impose relative degree or open-loop stability assumptions, or require domain-wide actuation. An adaptive design for our benchmark plant is a necessary step towards developing controllers for physical systems such as fluid, thermal, and chemical dynamics, where actuation can be only applied non-intrusively, the dynamics are unstable, and the parameters, such as the Reynolds, Rayleigh, Prandtl, or Peclet numbers are unknown because they vary with operating conditions. Our method builds upon our explicitly parametrized control formula in [26] to avoid solving a Riccati or Bezout equation at each time step.