# Galerkin approximation for inverse problems for nonautonomous nonlinear distributed systems

- Accepted:

DOI: 10.1007/BF01447744

- Cite this article as:
- Banks, H.T., Reich, S. & Rosen, I.G. Appl Math Optim (1991) 24: 233. doi:10.1007/BF01447744

## Abstract

We develop an abstract framework and convergence theory for Galerkin approximation for inverse problems involving the identification of nonautonomous, in general nonlinear, distributed parameter systems. We provide a set of relatively easily verified conditions which are sufficient to guarantee the existence of optimal solutions and their approximation by a sequence of solutions to a sequence of approximating finite-dimensional identification problems. Our approach is based upon the theory of monotone operators in Banach spaces and is applicable to a reasonably broad class of nonlinear distributed systems. Operator theoretic and variational techniques are used to establish a fundamental convergence result. An example involving evolution systems with dynamics described by nonstationary quasi-linear elliptic operators along with some applications and numerical results are presented and discussed.