Abstract. 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. Key words. on-line estimation, adaptive identification, parameter convergence, persistence of excitation, distributed parameter systems, infinite-dimensional systems, finite-dimensional approximation AMS subject classifications. 93B30, 93C25, 93C20, 65J10 PII. S0363012994270928 1. Introduction. In this paper we develop an abstract framework for the online, or adaptive, identification of unknown parameters for a class of infinite-dimensional and, in general, nonlinear dynamical systems. The estimator we construct takes the form of an infinite-dimensional linear evolution system with time-varying coefficients whose states consist of an estimator for the state of the plant and an estimator for the unknown parameters. Our scheme can estimate both constant and functional (e.g., spatially varying) parameters including the nonlinearity itself. That is, both the state space of the plant and the parameter space may be infinite dimensional. The results reported here were in fact obtained independently by two separate groups of researchers. The efforts of the first two authors (Baumeister and Scondo) culminated in the Ph.D. thesis of Dr. Scondo [30] , while the investigation by the second two authors (Demetriou and Rosen) led to the Ph.D. thesis of Dr. Demetriou [6] . In this paper we have attempted to capture the essence of the problem and its solution as treated independently by both groups of researchers in a clear and coherent manner. However, it should be noted that, as would be expected, there are some variations between the two treatments. Thus, the interested reader may also wish to consult the two theses, [6] and [30], in addition to the study that we present here.
Adaptive control schemes that rely on the estimation system states and unknown parameters of a plant were proposed by Baumeister et al. REF .
On-Line Parameter Estimation For Infinite-Dimensional Dynamical Systems
Siam Journal on Control and Optimization
Mathematics