An Approximation Theory for the Identification of Nonlinear Distributed Parameter Systems

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Abstract
An abstract approximation framework for the identification of nonlinear, distributed parameter systems is developed. Inverse problems for nonlinear systems governed by strongly maximal monotone operators (satisfying a mild continuous dependence condition with respect to the unknown parameters to be identified) are treated. Convergence of Galerkin approximations and the corresponding solutions of finite-dimensional approximating identification problems to a solution of the original infinite-dimensional identification problem is demonstrated, using the theory of nonlinear evolution systems and a nonlinear analogue of the Trotter–Kato approximation result for semigroups of bounded linear operators. The nonlinear theory developed here is shown to subsume an existing linear theory as a special case. It is also shown to be applicable to a broad class of nonlinear elliptic operators and the corresponding nonlinear parabolic partial differential equations to which they lead. An application of the theory to a quasilinear model for heat conduction or mass transfer is discussed.