Exact linear lumping in abstract spaces

Abstract

Exact linear lumping has earlier been defined for a finite dimensional space, that is, for the system of ordinary differential equations \(y'=f\circ y\) as a linear transformation \(M\) for which there exists a function \(\hat f\) such that \({\hat y}:=My\) itself obeys a differential equation \({\hat y}'={\hat f}\circ {\hat y}\). Here we extend the idea for the case when the values of \(y\) are taken in a Banach space. The investigations are restricted to the case when \(f\) is linear. Many theorems hold for the generalization of exact lumping, such as necessary and sufficient conditions for lumpability, and relations between the qualitative properties of the original and the transformed equations. The motivation behind the generalization of exact lumping is to apply the theory to reaction-diffusion systems, to an infinite number of chemical species, to continuous components, or to stochastic models as well.