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Antoine Chaillet

Publications and source records attributed to Antoine Chaillet.

At least 19 recordsLinked to original sources
arXiv2026

A Mathematical Characterization of Neural Activation Induced by Temporal Interference Stimulation

Temporal Interference Stimulation (TIS) is a non-invasive neuromodulation technique in which two high-frequency sinusoidal currents with slightly different frequencies generate a low-frequency envelope that can activate deep neural structures. This study investigates the conditions under which TIS elicits action potentials in a single neuron modeled by the FitzHugh-Nagumo system. This research integrates phase-plane analysis and geometric singular perturbation to develop a mathematical framework for analyzing TIS. By combining a mathematical analysis of differential equations with computer simulations, the study elucidates how the amplitudes and beat frequency jointly determine whether the neuron remains quiescent, exhibits only transient responses, or undergoes persistent (tonic) firing.

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arXiv2026

State-space fading memory

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 ($\delta$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 ($\delta$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 a current-driven memristive device possesses the FM property.

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arXiv2026

A superposition approach for the ISS Lyapunov-Krasovskii theorem with pointwise dissipation

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.

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arXiv2025

Disruption of parkinsonian brain oscillations

Deep brain stimulation (DBS) is an advanced surgical treatment for the symptoms of Parkinson's disease (PD), involving electrical stimulation of neurons within the basal ganglia region of the brain. DBS is traditionally delivered in an open-loop manner using fixed stimulation parameters, which may lead to suboptimal results. In an effort to overcome these limitations, closed loop DBS, using pathological subthalamic beta (13--30 Hz) activity as a feedback signal, offers the potential to adapt DBS automatically in response to changes in patient symptoms and side effects. However, clinically implemented closed-loop techniques have been limited to date to simple control algorithms, due to the inherent uncertainties in the dynamics involved. Model-free control, which has already seen successful applications in the field of bioengineering, offers a way to avoid this limitation and provides an alternative method to apply modern control approach to selective suppression of pathological oscillations.

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arXiv2025

Robust output stability and input-to-output stability based on output dissipation

LaSalle techniques to ensure the convergence of a given output usually fail at guaranteeing uniform convergence time, which induces robustness issues. Recent works have provided extra conditions under which a Lyapunov function that dissipates in terms of only the output guarantees this uniformity. In this paper, we extend these results to systems with inputs in order to establish either robust stability or input-to-output stability. In addition, we show that a recent relaxation of Barb__lat's lemma is also applicable to systems with inputs. The significance of the proposed results is demonstrated in the context of a recent adaptive control scheme.

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arXiv2025

Global Stabilization of Chemostats with Nonzero Mortality and Substrate Dynamics

In "chemostat"-type population models that incorporate substrate (nutrient) dynamics, the dependence of the birth (or growth) rate on the substrate concentration introduces nonlinear coupling that creates a challenge for stabilization that is global, namely, for all positive concentrations of the biomass and nutrients. This challenge for global stabilization has been overcome in the literature using relatively simple feedback when natural mortality of the biomass is absent. However, under natural mortality, it takes fortified, more complex feedback, outside of the existing nonlinear control design toolbox, to avoid biomass extinction from nutrient-depleted initial conditions. Such fortified feedback, the associated control Laypunov function design, and Lyapunov analysis of global stability are provided in this paper. We achieve global stabilization for two different chemostat models: (i) a lumped model, with two state variables, and (ii) a three-state model derived from an age-structured infinite-dimensional model. The proposed feedback stabilizers are explicit, applicable to both the lumped and the age-structured models, and coincide with simple feedback laws proposed in the literature when the mortality rate is zero. Global stabilization means subject to constraints: all positive biomass and nutrient concentrations are within the region of attraction of the desired equilibrium, and, additionally, this is achieved with a dilution input that is guaranteed to remain positive. For the lumped case with Haldane kinetics, we show that the reproduction rate dominating the mortality (excluding the reproduction and mortality being in balance) is not only sufficient but also necessary for global stabilization. The obtained results are illustrated with simple examples.

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arXiv2024

Adaptive observer and control of spatiotemporal delayed neural fields

An adaptive observer is proposed to estimate the synaptic distribution between neurons asymptotically from the measurement of a part of the neuronal activity and a delayed neural field evolution model. The convergence of the observer is proved under a persistency of excitation condition. Then, the observer is used to derive a feedback law ensuring asymptotic stabilization of the neural fields. Finally, the feedback law is modified to ensure simultaneously practical stabilization of the neural fields and asymptotic convergence of the observer under additional restrictions on the system. Numerical simulations confirm the relevance of the approach.

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arXiv2024

Forward completeness implies bounded reachable sets for time-delay systems on the state space of essentially bounded measurable functions

We consider time-delay systems with a finite number of delays in the state space $L^\infty\times\mathbb{R}^n$. In this framework, we show that forward completeness implies the bounded reachability sets property, while this implication was recently shown by J.L. Mancilla-Aguilar and H. Haimovich to fail in the state space of continuous functions. As a consequence, we show that global asymptotic stability is always uniform in the state space $L^\infty\times\mathbb{R}^n$.

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arXiv2023

Existence of an equilibrium for delayed neural fields under output proportional feedback

Recently, [2] proved that the closed-loop system resulting from the output proportional feedback stabilization of a class of delayed neural fields is input-to-state stable (ISS) for sufficiently high gain, subject to the existence of an equilibrium point for the closed-loop system. In the present paper, we show that a sufficient condition for such an equilibrium to exist is that the activation functions are bounded.

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arXiv2022

Online estimation of Hilbert-Schmidt operators and application to kernel reconstruction of neural fields

An adaptive observer is designed for online estimation of Hilbert-Schmidt operators from online measurement of the state for some class of nonlinear infinite-dimensional dynamical systems. Convergence is ensured under detectability and persistency of excitation assumptions. The class of systems considered is motivated by an application to kernel reconstruction of neural fields, commonly used to model spatiotemporal activity of neuronal populations. Numerical simulations confirm the relevance of the approach.

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arXiv2022

The ISS framework for time-delay systems: a survey

At the occasion of Eduardo D. Sontag's 70 th birthday, we provide here an overview of the tools available to study input-to-state stability (ISS) and related notions for time-delay systems. After a hopefully pedagogical presentation of the main differences with respect to the finite-dimensional theory, we review basic stability concepts for input-free time-delay systems, as well as instruments to guarantee them in practice, including the Lyapunov-Krasosvkii, Lyapunov-Razumikhin, and Halanay approaches. We then consider the influence of inputs through the notions of ISS, integral ISS, and input-to-output stability and provide both Lyapunov-like and solutions-based characterizations of these properties. We also show how these notions can be helpful for the stability analysis of interconnected systems, whether in cascade or in feedback form. We finally provide a list of questions which remain open until now.

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arXiv2022

Growth conditions for global exponential stability and exp-ISS of time-delay systems under point-wise dissipation

For time-delay systems, it is known that global asymptotic stability is guaranteed by the existence of a Lyapunov-Krasovskii functional that dissipates in a point-wise manner along solutions, namely whose dissipation rate involves only the current value of the solution's norm. So far, the extension of this result to global exponential stability (GES) holds only for systems ruled by a globally Lipschitz vector field and remains largely open for the input-to-state stability (ISS) property. In this paper, we rely on the notion of exponential ISS to extend the class of systems for which GES or ISS can be concluded from a point-wise dissipation. Our results in turn show that these properties still hold in the presence of a sufficiently small additional term involving the whole state history norm. We provide explicit estimates of the tolerable magnitude of this extra term and show through an example how it can be used to assess robustness with respect to modeling uncertainties.

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arXiv2022

Is Global Asymptotic Stability Necessarily Uniform for Time-Delay Systems?

For time-invariant finite-dimensional systems, it is known that global asymptotic stability (GAS) is equivalent to uniform global asymptotic stability (UGAS), in which the decay rate and transient overshoot of solutions are requested to be uniform on bounded sets of initial states. This paper investigates this relationship for time-invariant delay systems. We show that UGAS and GAS are equivalent for this class of systems under the assumption of robust forward completeness, i.e. under the assumption that the reachable set from any bounded set of initial states on any finite time horizon is bounded. We also show that, if the state space is a space in a particular family of Sobolev or Holder spaces, then GAS is equivalent to UGAS and that robust forward completeness holds. Based on these equivalences, we provide a novel Lyapunov characterization of GAS (and UGAS) in the aforementioned spaces.

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arXiv2020

Lyapunov Conditions for Uniform Asymptotic Output Stability and a Relaxation of Barbalat's Lemma

Asymptotic output stability (AOS) is an interesting property when addressing control applications in which not all state variables are requested to converge to the origin. AOS is often established by invoking classical tools such as Barbashin-Krasovskii-LaSalle's invariance principle or Barbalat's lemma. Nevertheless, none of these tools allow to predict whether the output convergence is uniform on bounded sets of initial conditions, which may lead to practical issues related to convergence speed and robustness. The contribution of this paper is twofold. First, we provide a testable sufficient condition under which this uniform convergence holds. Second, we provide an extension of Barbalat's lemma, which relaxes the uniform continuity requirement. Both these results are first stated in a finite-dimensional context and then extended to infinite-dimensional systems. We provide academic examples to illustrate the usefulness of these results and show that they can be invoked to establish uniform AOS for systems under adaptive control.

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arXiv2016

Global stabilization of linear systems with bounds on the feedback and its successive derivatives

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.

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arXiv2015

Global stabilization of classes of linear control systems with bounds on the feedback and its successive derivatives

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.

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arXiv2015

Global stabilization of multiple integrators by a bounded feedback with constraints on its successive derivatives

In this paper, we address the global stabilization of chains of integrators by means of a bounded static feedback law whose p first time derivatives are bounded. Our construction is based on the technique of nested saturations introduced by Teel. We show that the control amplitude and the maximum value of its p first derivatives can be imposed below any prescribed values. Our results are illustrated by the stabilization of the third order integrator on the feedback and its first two derivatives.

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