Approximation in the identification of second order degenerate distributed parameter systems

An approximation and convergence theory for off-line and adaptive identification of nonlinear, degenerate, second order distributed parameter systems is developed. The nonlinearity appears in the damping operator in the abstract wave equation. The degeneracy results from the possible noncoercivity of the system's leading operator coefficient which is assumed to be linear, symmetric and nonnegative. The stiffness operator is also assumed to be linear and symmetric.;The off-line approximation framework is based on a quotient space formulation for infinite dimensional dynamical systems and an abstract approximation result in the spirit of the Trotter-Kato Theorem. A sequence of approximating identification problems involving finite dimensional degenerate second order systems is considered and associated convergence aspects are addressed. The resulting theory is illustrated by an inverse problem for degenerate systems whose second order dynamics involve monotone operators and generalized Galerkin schemes with nonconforming elements.;The on-line or adaptive estimation framework involves the construction of a combined state and parameter estimator as an infinite dimensional, non-degenerate, second order system. Asymptotic state convergence is established by a Lyapunov-like estimate. Parameter convergence is proved under an additional assumption of persistence of excitation. A finite dimensional approximation theory is developed. Examples and relevant numerical studies are presented.