Antoine Chaillet

Gustave Bainier, Antoine Chaillet, Rodolphe Sepulchre et al.

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.

Jonathan Laporte, Antoine Chaillet, Yacine Chitour

We address the global stabilization of linear time-invariant (LTI) systems when the magnitude of the control input and its successive time derivatives, up to an order $p\in\mathbb N$, are bounded by prescribed values. We propose a static state feedback that solves this problem for any admissible LTI systems, namely for stabilizable systems whose internal dynamics has no eigenvalue with positive real part. This generalizes previous work done for single-input chains of integrators and rotating dynamics.

Andrii Mironchenko, Fabian Wirth, Antoine Chaillet et al.

We show that the existence of a Lyapunov-Krasovskii functional (LKF) with pointwise dissipation (i.e. dissipation in terms of the current solution norm) suffices for input-to-state stability, provided that uniform global stability can also be ensured using the same LKF. To this end, we develop a stability theory, in which the behavior of solutions is not assessed through the classical norm but rather through a specific LKF, which may provide significantly tighter estimates. We discuss the advantages of our approach by means of an example.

Jonathan Laporte, Antoine Chaillet, Yacine Chitour

In this paper, we address the problem of globally stabilizing a linear time-invariant (LTI) system by means of a static feedback law whose amplitude and successive time derivatives, up to a prescribed order $p$, are bounded by arbitrary prescribed values. We solve this problem for two classes of LTI systems, namely integrator chains and skew-symmetric systems with single input. For the integrator chains, the solution we propose is based on the nested saturations introduced by A.R. Teel. We show that this construction fails for skew-symmetric systems and propose an alternative feedback law. We illustrate these findings by the stabilization of the third order integrator with prescribed bounds on the feedback and its first two derivatives, and similarly for the harmonic oscillator with prescribed bounds on the feedback and its first derivative.