On the regularity of lumped nonlinear dynamics in Banach spaces

Fatihcan M. Atay and Lavinia Roncoroni

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Submission date: 07. Oct. 2015
Pages: 17
Bibtex
MSC-Numbers: 46Txx, 47Hxx, 47Axx, 37C75
Keywords and phrases: reduction, lumping, aggregation, differentiability, linearization, stability theory
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Abstract:
We study the regularity properties of the lumping problem for differential equations in Banach spaces, namely the projection of dynamics by a reduction operator onto a reduced state space in which a self-contained dynamical description exists. We study dynamics generated by a nonlinear operator F and a linear and bounded reduction operator M. We first show, using quotient space methods, that the reduced operator is C¹, provided that F itself is C¹ in the original state space. We further prove that a particular lumping relation holds between the Fréchet differentials of F and the reduced operator. In this way, by smoothness, the linearization principle applies and it is possible to use results from linear theory to study the local behavior of the system.