State-space fading memory

A team of researchers including Gustave Bainier, Antoine Chaillet, Rodolphe Sepulchre, and Alessio Franci has published a significant theoretical paper titled 'State-space fading memory' on arXiv. The work addresses a fundamental gap in systems theory by formally connecting the fading memory (FM) property—which describes how past inputs gradually lose influence on current outputs—with state-space notions of nonlinear stability, particularly incremental stability. The authors introduce a precise state-space definition of FM, interpreting it as an extension of incremental input-to-output stability (δIOS) that explicitly incorporates a mathematical kernel bounding the decay of past input differences.

Crucially, the paper demonstrates that incremental input-to-state stability (δISS) implies the fading memory property semi-globally for time-invariant systems, given an equibounded input assumption. This theoretical bridge is important because it means the powerful approximation theorems originally developed by Boyd and Chua—which show that fading memory systems can be uniformly approximated by finite-dimensional models—now apply directly to δISS state-space models. As a practical application, the authors show that under mild assumptions, the state-space models of current-driven memristors possess the FM property. This work provides a more rigorous mathematical foundation for analyzing and designing systems where memory and stability interact, such as in reservoir computing and recurrent neural networks (RNNs).