Adaptive Parameter Estimation for Degenerate Parabolic Systems

We consider the adaptive, or on-line, estimation of parameters in implicit and possibly degenerate parabolic systems. A combined state and parameter estimator is constructed as an initial value problem for an infinite dimensional non-degenerate evolution equation in weak or variational form. State convergence is established via a Lyapunov-like estimate. The finite dimensional notion of persistence of excitation is extended to the implicit infinite dimensional case and used to establish parameter convergence. A finite dimensional approximation theory is developed. Results of our numerical studies on a degenerate one dimensional heat equation are presented to demonstrate the feasibility of our approach.