Yacine Chitour

Université Paris-Saclay · physics

In this study, we investigate the intricate connection between visual perception and the mathematical modelling of neural activity in the primary visual cortex (V1), focusing on replicating the MacKay effect [Mackay, Nature 1957]. While bifurcation theory has been a prominent mathematical approach for addressing issues in neuroscience, especially i...

To study the interaction between retinal stimulation by redundant geometrical patterns and the cortical response in the primary visual cortex (\({{\,\textrm{V1}\,}}\)), we focus on the MacKay effect (Nature, 1957) and Billock and Tsou’s experiments (PNAS, 2007). We use a controllability approach to describe these phenomena starting from a classical...

We establish the exponential decay of the solutions of the damped wave equations in one-dimensional space where the damping coefficient is a nowhere-vanishing function of space. The considered PDE is associated with several dynamic boundary conditions, also referred to as Wentzell/Ventzel boundary conditions in the literature. The analysis is based...

This paper concerns some spectral properties of the scalar dynamical system defined by a linear delay-differential equation with two positive delays. More precisely, the existing links between the delays and the maximal multiplicity of the characteristic roots are explored, as well as the dominance of such roots compared with the spectrum localizat...

Consider the saturated complex double integrator, i.e., the linear control system \(\dot{x}=Ax+B\sigma (u)\), where \(x\in {\mathbb {R}}^4\), \(u\in {\mathbb {R}}\), \(B\in {\mathbb {R}}^4\), the \(4\times 4\) matrix A is not diagonalizable and admits a nonzero purely imaginary eigenvalue of multiplicity two, the pair (A, B) is controllable and \(\...

We establish the decay of the solutions of the damped wave equations in one dimensional space for the Dirichlet, Neumann, and dynamic boundary conditions where the damping coefficient is a function of space and time. The analysis is based on the study of the corresponding hyperbolic systems associated with the Riemann invariants. The key ingredient...

To study the interaction between retinal stimulation by redundant geometrical patterns and the cortical response in the primary visual cortex (V1), we focus on the MacKay effect (Nature, 1957) and Billock and Tsou's experiments (PNAS, 2007). Starting from a classical biological model of neuronal fields equations with a non-linear response function,...

The paper deals with the controllability of finite-dimensional linear difference delay equations, i.e., dynamics for which the state at a given time $t$ is obtained as a linear combination of the control evaluated at time $t$ and of the state evaluated at finitely many previous instants of time $t-\Lambda_1,\dots,t-\Lambda_N$. Based on the realizat...

In this paper we discuss the notion of universality for classes of candidate common Lyapunov functions of linear switched systems. On the one hand, we prove that a family of absolutely homogeneous functions is universal as soon as it approximates arbitrarily well every convex absolutely homogeneous function for the $C^0$ topology of the unit sphere...

In this paper, we study the $L^p$-asymptotic stability with $p\in (1,\infty)$ of the one-dimensional nonlinear damped wave equation with a localized damping and Dirichlet boundary conditions in a bounded domain $(0,1)$. We start by addressing the well-posedness problem. We prove the existence and the uniqueness of weak solutions for $p\in [2,\infty...

Understanding sensory-induced cortical patterns in the primary visual cortex V1 is an important challenge both for physiological motivations and for improving our understanding of human perception and visual organisation. In this work, we focus on pattern formation in the visual cortex when the cortical activity is driven by a geometric visual hall...

In this paper, two classes of continuous higher order adaptive sliding mode controllers based on barrier functions are developed for a perturbed chain of integrators with unbounded perturbations. Both classes provide finite-time convergence of system states to a predefined domain using a continuous control signal. The first class of adaptive contro...

In this paper we consider the problem of determining the stability properties, and in particular assessing the exponential stability, of a singularly perturbed linear switching system. One of the challenges of this problem arises from the intricate interplay between the small parameter of singular perturbation and the rate of switching, as both ten...

This paper concerns some spectral properties of the scalar dynamical system defined by a linear delay-differential equation with two positive delays. More precisely, the existing links between the delays and the maximal multiplicity of the characteristic roots are explored, as well as the dominancy of such roots compared with the spectrum localizat...

In this paper, we propose a geometric framework to analyze the convergence properties of gradient descent trajectories in the context of linear neural networks. We translate a well-known empirical observation of linear neural nets into a conjecture that we call the overfitting conjecture which states that, for almost all training data and initial c...

This paper addresses the problem of estimating the tilt and more generally the attitude of a rigid body that is subject to high accelerations and equipped with inertial measurement units (IMU) and a sensor providing the body velocity (expressed in the reference frame attached to the body). In the absence of a magnetometer, tilt estimation is propos...

Consider a non-autonomous continuous-time linear system in which the time-dependent matrix determining the dynamics is piecewise constant and takes finitely many values $A_1, \dotsc, A_N$. This paper studies the equality cases between the maximal Lyapunov exponent associated with the set of matrices $\{A_1, \dotsc, A_N\}$, on the one hand, and the...

p>In this paper, we study the L^p-asymptotic stability of the one dimensional linear damped wave equation with Dirichlet boundary conditions in [ 0 , 1 ] , with p ∈ ( 1 , ∞ ) . The damping term is assumed to be linear and localized to an arbitrary open sub-interval of [ 0 , 1 ] . We prove that the semi-group S p ( t ) t ≥ 0 associated wi...

This paper is the first of two parts which considers the rolling (or development) of two Riemannian connected manifolds (M,g) and \(\left (\hat {M},\hat {g}\right )\) of dimensions 2 and 3 respectively, with the constraints of no-spinning and no-slipping. The present work is a continuation of Mortada et al. (Acta Appl Math 139:105–31, 2015), which...

It has been observed in recent works that, for several classes of linear time-invariant time-delay systems of retarded or neutral type with a single delay, if a root of its characteristic equation attains its maximal multiplicity, then this root is the rightmost spectral value, and hence it determines the exponential behavior of the system, a prope...

It has been shown in several recent works that, for some classes of linear time-delay systems, spectral values of maximal multiplicity are dominant, a property known as multiplicity-induced-dominancy (MID). This paper starts the investigation of whether MID holds for delay differential-algebraic systems by considering a single-delay system composed...

This paper is the second part of a study initiated by Mortada et al. (2020), where we have considered the rolling (or development) of two Riemannian connected manifolds (M,g) and (M̂,ĝ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepack...

This paper is concerned with the analysis of a 1D wave equation $z_{tt}-z_{xx}=0$ on $[0,1]$ with a Dirichlet condition at $x=0$ and a damping acting at $x=1$ of the form $(z_t(t,1),-z_x(t,1))\in\Sigma$ for $t\geq 0$, where $\Sigma$ is a given subset of $\mathbb R^2$. The study is performed within an $L^p$ functional framework, $p\in [1, +\infty]$....

In this paper we deal with infinite-dimensional nonlinear forward complete dynamical systems which are subject to uncertainties. We first extend the well-known Datko lemma to the framework of the considered class of systems. Thanks to this generalization, we provide characterizations of the uniform (with respect to uncertainties) local, semi-global...

We study asymptotic stability of continuous-time systems with mode-dependent guaranteed dwell time. These systems are reformulated as special cases of a general class of mixed (discrete–continuous) linear switching systems on graphs, in which some modes correspond to discrete actions and some others correspond to continuous-time evolutions. Each di...

Consider the saturated complex double integrator, i.e., the linear control system $\dot x=Ax+B\sigma(u)$, where $x\in\R^4$, $u\in\R$, $B\in\R^4$, the $4\times 4$ matrix $A$ is not diagolizable and admits a non zero purely imaginary eigenvalue of multiplicity two, the pair $(A,B)$ is controllable and $\sigma:\R\to\R$ is a saturation function. We pro...

In this paper, we study the $L^p$-asymptotic stability of the one-dimensional linear damped wave equation with Dirichlet boundary conditions in $[0,1]$, with $p\in (1,\infty)$. The damping term is assumed to be linear and localized to an arbitrary open sub-interval of $[0,1]$. We prove that the semi-group $(S_p(t))_{t\geq 0}$ associated with the pr...

In this paper, we present barrier function-based adaptive controllers for fast stabilization of a perturbed chain of integrators with bounded uncertainties. We refer to such controllers as adaptive higher order sliding mode controllers since they are designed for a perturbed chain of integrators of length r with bounded uncertainties such that the...

It has been observed in recent works that, for several classes of linear time-invariant time-delay systems of retarded or neutral type with a single delay, if a root of its characteristic equation attains its maximal multiplicity, then this root is the rightmost spectral value, and hence it determines the exponential behavior of the system, a prope...

Given a discrete-time linear switched system Σ(A) associated with a finite set A of matrices, we consider the measures of its asymptotic behavior given by, on the one hand, its deterministic joint spectral radius ρd(A) and, on the other hand, its probabilistic joint spectral radii ρp(ν,P,A) for Markov random switching signals with transition matrix...

In this paper, a variable gain super-twisting algorithm based on a barrier function is proposed for a class of first order disturbed systems with uncertain control coefficient and whose disturbances derivatives are bounded but the upper bounds of those derivatives are unknown. The specific feature of this algorithm is that it can ensure the converg...

This paper is concerned with the asymptotic stability analysis of a one dimensional wave equation with Dirichlet boundary conditions subject to a nonlinear distributed damping within an Lp functional framework, p∈[2,∞]. Some well-posedness results are provided together with exponential decay to zero of trajectories, with an estimation of the decay...

In this paper, we present an observation scheme, with proven Lyapunov stability, for estimating a humanoid's floating base orientation. The idea is to use velocity aided attitude estimation, which requires to know the velocity of the system. This velocity can be obtained by taking into account the kinematic data provided by contact information with...

In this paper, we present an observation scheme, with proven Lyapunov stability, for estimating a humanoid's floating base orientation. The idea is to use velocity aided attitude estimation, which requires to know the velocity of the system. This velocity can be obtained by taking into account the kinematic data provided by contact information with...

In this paper we estimate the worst rate of exponential decay of degenerate gradient flows $\dot x = -S x$, issued from adaptive control theory. Under persistent excitation assumptions on the positive semi-definite matrix $S$, we provide upper bounds for this rate of decay consistent with previously known lower bounds and analogous stability result...

In this paper, we study input-to-state (ISS) issues for damped wave equations with Dirichlet boundary conditions on a bounded domain of dimension two. The damping term is assumed to be non-linear and localized to an open subset of the domain. In a first step, we handle the undisturbed case as an extension of a previous work, where stability results...

This paper addresses the problem of estimating the attitude of a rigid body, which is subject to high accelerations and equipped with inertial measurement unit (IMU) and sensors providing the body velocity (expressed in the reference frame attached to the body). That issue can be treated differently depending on the level of confidence in the measu...

In this article, we consider the rolling (or development) of two Riemannian connected manifolds $(M,g)$ and $(\hat{M},\hat{g})$ of dimensions $2$ and $3$ respectively, with the constraints of no-spinning and no-slipping. The present work is a continuation of \cite{MortadaKokkonenChitour}, which modelled the general setting of the rolling of two Rie...

In this paper, we present Lyapunov-based {\color{black}time varying} controllers for {\color{black}fast} stabilization of a perturbed chain of integrators with bounded uncertainties. We refer to such controllers as {\color{black}time varying} higher order sliding mode controllers since they are designed for nonlinear Single-Input-Single-Output (SIS...

In this paper we deal with infinite-dimensional nonlinear forward complete dynamical systems which are subject to external disturbances. We first extend the well-known Datko lemma to the framework of the considered class of systems. Thanks to this generalization, we provide characterizations of the uniform (with respect to disturbances) local, semi...

It has been observed in several recent works that, for some classes of linear time-delay systems, spectral values of maximal multiplicity are dominant, a property known as multiplicity-induced-dominancy (MID). This paper starts the investigation of whether MID holds for delay differential-algebraic systems by considering a single-delay system compo...

In this paper, we study approximate and exact controllability of the linear difference equation $x(t) = \sum_{j=1}^N A_j x(t - \Lambda_j) + B u(t)$ in $L^2$, with $x(t) \in \mathbb C^d$ and $u(t) \in \mathbb C^m$, using as a basic tool a representation formula for its solution in terms of the initial condition, the control $u$, and some suitable ma...

In this paper, we study the worst rate of exponential decay for degenerate gradient flows in ℝⁿ of the form ẋ(t) = —c(t)c(t)Tx(t), issued from adaptative control theory, under a persistent excitation (PE) condition. That is, there exists a, b, T > 0 such that, for every t ≥ 0 it holds aIdn ≤ ∫t+Tt c(s)c(s)T ds ≤ bIdn. Our main result is an upper bo...

We study asymptotic stability of continuous-time systems with mode-dependent guaranteed dwell time. These systems are reformulated as special cases of a general class of mixed (discrete-continuous) linear switching systems on graphs, in which some modes correspond to discrete actions and some others correspond to continuous-time evolutions. Each di...

In this paper, we study the indirect boundary stability and exact controllability of a one-dimensional Timoshenko system. In the first part of the paper, we consider the Timoshenko system with only one boundary fractional damping. We first show that the system is strongly stable but not uniformly stable. Hence, we look for a polynomial decay rate f...

In this paper, a variable gain super-twisting algorithm based on a barrier function is proposed for a class of first order disturbed systems with uncertain control coefficient and whose disturbances derivatives are bounded but they are unknown. The specific feature of this algorithm is that it can ensure the convergence of the output variable and m...

In this article we study how bad can be the singularities of a time-optimal trajectory of a generic control affine system. In the case where the control is scalar and belongs to a closed interval it was recently shown in [6] that singularities cannot be, generically, worse than finite order accumulations of Fuller points, with order of accumulation...

In this paper, by means of the Riesz basis approach, we study the stability of a weakly damped system of two second order evolution equations coupled through the velocities (see (1.1)). If the fractional order damping becomes viscous and the waves propagate with equal speeds, we prove exponential stability of the system and, otherwise, we establish...

In this article, we provide a general strategy based on Lyapunov functionals to analyse global asymptotic stability of linear infinite-dimensional systems subject to nonlinear dampings under the assumption that the origin of the system is globally asymptotically stable with a $linear\ damping$ . To do so, we use the fact that, for any linear infi...

In this paper, we consider issues relative to prescribed time stabilisation of a chain of integrators of arbitrary length, either pure (i.e., where there is no disturbance) or perturbed. In a first part, we revisit the proportional navigation feedback (PNF) approach and we show that it can be appropriately recasted within the framework of time-vary...

This paper is concerned with the asymptotic stability analysis of a one dimensional wave equation with Dirichlet boundary conditions subject to a nonlinear distributed damping with an L p functional framework, p ∈ [2, ∞]. Some well-posedness results are provided together with exponential decay to zero of trajectories, with an estimation of the deca...

Over the past decade, many medical lower limb exoskeletons have been developed and exploited. The advantage of such a systems is to ensure the mobility of paraplegic patients, as well as their physical rehabilitation. However, existing solutions have not been widely available among the disabled population, particularly adolescents, due to the limit...

In this paper, we study the asymptotic growth of the eigenvalues of the Laplace-Beltrami operator on singular Riemannian manifolds, where all geometrical invariants appearing in classical spectral asymptotics are unbounded, and the total volume can be infinite. Under suitable assumptions on the curvature blow-up, we show how the singularity influen...

This paper is concerned with the asymptotic stability analysis of a one dimensional wave equation subject to a nonmonotone distributed damping. A well-posedness result is provided together with a precise characterization of the asymptotic behavior of the trajectories of the system under consideration. The well-posedness is proved in the nonstandard...

In this paper, we study the indirect boundary stability and exact controllability of a one-dimensional Timoshenko system. In the first part of the paper, we consider the Timoshenko system with only one boundary fractional damping. We first show that the system is strongly stable but not uniformly stable. Hence, we look for a polynomial decay rate f...

This paper is concerned with the asymptotic stability analysis of a one dimensional wave equation subject to a nonmonotone distributed damping. A well-posedness result is provided together with a precise characterization of the asymptotic behavior of the trajectories of the system under consideration. The well-posedness is proved in the nonstandard...

Given a discrete-time linear switched system $\Sigma(\mathcal A)$ associated with a finite set $\mathcal A$ of matrices, we consider the measures of its asymptotic behavior given by, on the one hand, its deterministic joint spectral radius $\rho_{\mathrm d}(\mathcal A)$ and, on the other hand, its probabilistic joint spectral radii $\rho_{\mathrm p...

In this article we present a geometric framework to analyze convergence of gradient descent trajectories in the context of neural networks. In the case of linear networks of an arbitrary number of hidden layers, we characterize appropriate quantities which are conserved along the gradient descent system (GDS). We use them to prove boundedness of ev...

The paper presents a new observer for tilt estimation of a 3-D non-rigid pendulum. The system can be seen as a multibody robot attached to the environment with a ball joint. There is no sensor for the joint position of the sensor. The estimation of tilt, i.e. roll and pitch angles, is mandatory for balance control for a humanoid robot and all tasks...

In this paper, by means of the Riesz basis approach, we study the stability of a weakly damped system of two second order evolution equations coupled through the velocities. If the fractional order damping becomes viscous and the waves propagate with equal speeds, we prove exponential stability of the system and, otherwise, we establish an optimal...

In this article, we provide a general strategy based on Lyapunov functionals to analyse global asymptotic stability of linear infinite-dimensional systems subject to nonlinear dampings under the assumption that the origin of the system is globally asymp-totically stable with a linear damping. To do so, we first characterize, in terms of Lyapunov fu...

We prove the $C^{1}$ regularity for a class of abnormal length-minimizers in rank $2$ sub-Riemannian structures. As a consequence of our result, all length-minimizers for rank $2$ sub-Riemannian structures of step up to $4$ are of class $C^{1}$.

This note presents a new adaptive attitude tracking controller for rigid body systems, with unknown inertia and unknown gyro-bias, using inertial vector measurements. The proposed control scheme guarantees almost global asymptotic convergence of the attitude and angular velocity to their desired values. Simulation results are provided to illustrate...

This paper brings an identified model for a 6 degrees of freedom (dof) industrial robot, the Denso VP-6242G robot, first without payload, then with a payload. This last is composed of a force sensor fixed between a spherical handle and the robot end-effector. This equipped end-effector is intended to experiments in the field of Physical Human-Robot...

- This article deals with the derivation of ISS-Lyapunov functions for infinite-dimensional linear systems subject to saturations. Two cases are considered: 1) the saturation acts in the same space as the control space; 2) the saturation acts in another space, especially a Banach space. For the first case, an explicit ISS-Lyapunov function can be d...

In this paper, we study approximate and exact controllability of the linear difference equation $x(t) = \sum\_{j=1}^N A\_j x(t - \Lambda\_j) + B u(t)$ in $L^2$, with $x(t) \in \mathbb C^d$ and $u(t) \in \mathbb C^m$, using as a basic tool a representation formula for its solution in terms of the initial condition, the control $u$, and some suitable...

In this paper, we address the question whether input-to-state stability (ISS) of nonlinear time-delay systems is guaranteed when a Lyapunov-Krasovskii functional (LKF) satisfies a dissipation inequality in which the dissipation rate involves solely the present value of the state. We do not yet confirm or infirm this conjecture, but rather identify...

In this paper we address the exponential stability of a system of transport equations with intermittent damping on a network of $N \geq 2$ circles intersecting at a single point $O$. The $N$ equations are coupled through a linear mixing of their values at $O$, described by a matrix $M$. The activity of the intermittent damping is determined by pers...

In this paper, we present a generalization of the supertwisting algorithm for perturbed chains of integrators of arbitrary order. This Higher Order Super-Twisting (HOST) controller is homogeneous with respect to a family of dilations and is continuous. It is built as a dynamic controller (with respect to the state variable of the chain of integrato...

In this paper, we address the stability of non-autonomous difference equations by providing an explicit formula expressing the solution at time $t$ in terms of the initial condition and time-dependent matrix coefficients. We then relate the asymptotic behavior of such coefficients to that of solutions. As a consequence, we obtain necessary and suff...

In this paper, we present a Lyapunov-based homogeneous controller for the stabilization of a perturbed chain of integrators of arbitrary order r ≥ 1. The proposed controller is based on homogeneous controller for stabilization of pure chain of integrators. The control of homogeneity degree is also introduced and various controllers are designed usi...

In this paper, we present Lyapunov-based adaptive controllers for the practical (or real) stabilization of a perturbed chain of integrators with bounded uncertainties. We refer to such controllers as Adaptive Higher Order Sliding Mode (AHOSM) controllers since they are designed for nonlinear SISO systems with bounded uncertainties such that the unc...

In this paper, we present the controllability properties of Keplerian motion controlled by low-thrust control systems. The low-thrust control system, compared with high or even impulsive control system, provide a fuel-efficient means to control the Keplerian motion of a satellite in restricted two-body problem. We obtain that, for any positive valu...

Second order systems whose drift is defined by the gradient of a given potential are considered, and minimization of the $L^1$-norm of the control is addressed. An analysis of the extremal flow emphasizes the role of singular trajectories of order two [25,29]; the case of the two-body potential is treated in detail. In $L^1$-minimization, regular e...