On-line Parameter Estimation for Infinite-dimensional Dynamical Systems∗

The on-line or adaptive identification of parameters in abstract linear and nonlinear infinite-dimensional dynamical systems is considered. An estimator in the form of an infinitedimensional linear evolution system having the state and parameter estimates as its states is defined. Convergence of the state estimator is established via a Lyapunov estimate. The finite-dimensional notion of a plant being sufficiently rich or persistently excited is extended to infinite dimensions. Convergence of the parameter estimates is established under the additional assumption that the plant is persistently excited. A finite-dimensional approximation theory is developed, and convergence results are established. Numerical results for examples involving the estimation of both constant and functional parameters in one-dimensional linear and nonlinear heat or diffusion equations and the estimation of stiffness and damping parameters in a one-dimensional wave equation with Kelvin–Voigt viscoelastic damping are presented.