State-space fading memory
This work addresses a theoretical gap in systems approximation and reservoir computing by linking fading memory to state-space stability, though it is incremental as it builds on existing operator-theoretic frameworks.
The paper tackles the understudied connection between fading memory and state-space stability by introducing a state-space definition of fading memory, showing that incremental input-to-state stability implies fading memory semi-globally for time-invariant systems, and applying this to prove that current-driven memristor models possess the fading memory property.
The fading-memory (FM) property captures the progressive loss of influence of past inputs on a system's current output and has originally been formalized by Boyd and Chua in an operator-theoretic framework. Despite its importance for systems approximation, reservoir computing, and recurrent neural networks, its connection with state-space notions of nonlinear stability, especially incremental ones, remains understudied. This paper introduces a state-space definition of FM. In state-space, FM can be interpreted as an extension of incremental input-to-output stability ($δ$IOS) that explicitly incorporates a memory kernel upper-bounding the decay of past input differences. It is also closely related to Boyd and Chua's FM definition, with the sole difference of requiring uniform, instead of general, continuity of the memory functional with respect to an input-fading norm. We demonstrate that incremental input-to-state stability ($δ$ISS) implies FM semi-globally for time-invariant systems under an equibounded input assumption. Notably, Boyd and Chua's approximation theorems apply to delta-ISS state-space models. As a closing application, we show that, under mild assumptions, the state-space model of current-driven memristors possess the FM property.