Inverse Problems for Nonlinear Delay Systems

Abstract

Abstract. We consider inverse or parameter estimation problems for general nonlinear nonautonomous dynamical systems with delays. The parameters may be from a Euclidean set as usual, may be time dependent coefficients or may be probability distributions across a population as arise in aggregate data problems. Theoretical convergence results for finite dimensional approximations to the systems are given. Several examples are used to illustrate the ideas and computational results that demonstrate efficacy of the approximations are presented.Key words. Nonlinear delay systems, inverse problems, uncertainty, computational methods.AMS subject classifications. 34C55, 34K06, 34K28, 34K29, 34G20, 93C10.1. Introduction. Delay differential equations have been a topic of much interest in the mathematical research literature for more than 50 years. Contributions range from classical applications and theoretical and computational methodologies [Ba79, Ba82, BBu1, BBu2, BKap, BellCook, Cushing, Diekmann, Driver, Gorecki, JKH1, JKH2, JKH3, Kap82, KapSal87, KapSal89, KapSch, Kuang, Minorsky, Webb, Wright] to modern applications in biology [BBJ, BBH, MSNP, NMiP, NMuP, NP]. In this paper we return to a topic that has become increasingly relevant in current research: a theoretical and computational approach for inverse problems involving nonlinear delay systems. One approach that is by now classical dates back to the 1970's [Ba79, BBu1, BBu2, BKap]. In this approach one approximates solutions to the infinite dimensional state systems such as (1) below by first converting them to an abstract evolution equation in a functional analytic state space setting. One can approximate solutions in finite dimensional subspaces spanned by pre-chosen basis elements (e.g., piece-wise linear or cubic splines) in a Galerkin approach which is equivalent to a finite element approximation framework (as is classically used for partial differential equations). One is then able to numerically calculate the generalized Fourier coefficients of approximate solutions relative to the splines, and with these coefficients, recover an approximation to the solutions of delay systems (1).Here we turn to the mathematical aspects of these nonlinear FDE systems and present an outline of the necessary mathematical and numerical analysis foundations. Thus we provide an extension (to treat time dependent coefficients and general parameters including probability measures) of arguments for approximation and convergence in inverse problems found in [Ba82].For nonlinear delay systems such as those discussed here, approximation in the context of a linear semigroup framework as presented [BBu1, BBu2, BKap] is not direct. However one can use the ideas of that theory as a basis for a wide class of nonlinear delay system approximations. Details in this direction can be found in the early work [Ba79, Kap82] which is a direct extension of the results of [BBu1, BBu2, BKap] to nonlinear delay systems. The new theoretical results presented here are extensions of these earlier ideas t...

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