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Systematization of Approaches to Adaptive Boundary Stabilization of PDEs

While adaptive control of finite dimensional systems is an advanced field that has produced adaptive control methods for a very general class of LTI systems, adaptive control techniques have been developed for only a few of the classes of PDEs for which non-adaptive controllers exist. We present a catalog of approaches for the design of adaptive controllers for PDEs controlled from a boundary and containing unknown destabilizing parameters affecting the interior of the domain. We differentiate between two major classes of schemes: Lyapunov schemes and certainty equivalence schemes. Within the certainty equivalence class, two types of identifier designs are pursued: passivity-based and swapping designs. Each of those designs is applicable to two types of parametrizations: the plant model in its original form (which we refer to as the "u-model") and a transformed model to which a backstepping transformation has been applied (which we refer to as the "w-model"). Hence, a large number of control algorithms result from combining different design tools - Lyapunov schemes, w-passive schemes, u-swapping schemes, etc. Our method builds upon the explicitly parametrized control formulae that we introduced in our earlier work on non-adaptive backstepping control for PDEs. These formulae allow us to develop tunable controllers that avoid solving Riccati or Bezout equations at each time step