arXiv subjects

Yacine Chitour

Publications and source records attributed to Yacine Chitour.

At least 37 records · Page 2Linked to original sources

arXiv2024

Intrinsic Derivation of the Equations of a Snake Robot based on a Cosserat Beam Model

In this paper, we present an intrinsic derivation of the equations ruling the dynamics motion of a snake robot dynamics. Based on a Cosserat beam model, we first show that the extended configuration space is a Lie group. Endowing it with an appropriate left invariant metric, the corresponding Euler-Poincaré equations can be reduced to a system of hyperbolic PDEs in the Lie algebra $\mathfrak{se}(3)$. We also provide the constitutive law describing the actuation in this system of PDEs.

math.OC↗

arXiv2024

Lyapunov functions for linear damped wave equations in one-dimensional space with dynamic boundary conditions

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 on the determination of appropriate Lyapunov functions and some further analysis. This result is associated with a regulation problem inspired by a real experiment with a proportional-integral control. Some numerical simulations and additional results on closed wave equations are also provided.

math.AP↗

arXiv2024

A corona theorem for an algebra of Radon measures with an application to exact controllability for linear controlled delayed difference equations

This paper proves a corona theorem for the algebra of Radon measures compactly supported in $\mathbb{R}_-$ and this result is applied to provide a necessary and sufficient Hautus--type frequency criterion for the $L^1$ exact controllability of linear controlled delayed difference equations (LCDDE). Hereby, it solves an open question raised in [6].

math.OC↗

arXiv2023

Weyl's law for singular Riemannian manifolds

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 influences the Weyl's asymptotics. Our main motivation comes from the construction of singular Riemannian metrics with prescribed non-classical Weyl's law. Namely, for any non-decreasing slowly varying function $\upsilon$ we construct a singular Riemannian structure whose spectrum is discrete and satisfies \[ N(λ) \sim \frac{ω_n}{(2π)^n} λ^{n/2} \upsilon(λ). \] Examples of slowly varying functions are $\logλ$, its iterations $\log_k λ= \log_{k-1}\logλ$, any rational function with positive coefficients of $\log_k λ$, and functions with non-lo\-ga\-rithmic growth such as $\exp\left((\log λ)^{α_1} \dots (\log_k λ)^{α_k} \right)$ for $α_i \in (0,1)$. A key tool in our arguments is a new quantitative estimate for the remainder of the heat trace and the Weyl's function on Riemannian manifolds, which is of independent interest.

math.DG↗

arXiv2023

Cortical origins of MacKay-type visual illusions: A case for the non-linearity

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). We use a controllability approach to describe these phenomena starting from a classical biological model of neuronal field equations with a non-linear response function. The external input containing a localised control function is interpreted as a cortical representation of the static visual stimuli used in these experiments. We prove that while the MacKay effect is essentially a linear phenomenon (i.e., the nonlinear nature of the activation does not play any role in its reproduction), the phenomena reported by Billock and Tsou are wholly nonlinear and depend strongly on the shape of the nonlinearity used to model the response function.

q-bio.NC↗

arXiv2023

On the pole placement of scalar linear delay systems with two delays

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 localization. As a by-product of the analysis, the pole placement issue is revisited with more emphasis on the role of the delays as control parameters in defining a partial pole placement guaranteeing the closed-loop stability with an appropriate decay rate of the corresponding dynamical system.

math.DS↗

arXiv2022

Exponential decay of solutions of damped wave equations in one dimensional space in the $L^p$ framework for various boundary conditions

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 in the study of these systems is the use of the internal dissipation energy to estimate the difference of solutions with their mean values in an average sense.

math.OC↗

arXiv2022

On the gap between deterministic and probabilistic Lyapunov exponents for continuous-time linear systems

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 corresponding ones for piecewise deterministic Markov processes with modes $A_1, \dotsc, A_N$, on the other hand. A fundamental step in this study consists in establishing a result of independent interest, namely, that any sequence of Markov processes associated with the matrices $A_1,\dotsc, A_N$ converges, up to extracting a subsequence, to a Markov process associated with a suitable convex combination of those matrices.

math.OC↗

arXiv2022

Lp-asymptotic stability of 1D damped wave equations with localized and nonlinear damping

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 $p\in [2,\infty)$ and the existence and the uniqueness of strong solutions for all $p\in [1,\infty)$. The proofs rely on the well-posedness already proved in the $L^\infty$ framework by [4] combined with a density argument. Then we prove that the energy of strong solutions decays exponentially to zero. The proof relies on the multiplier method combined with the work that has been done in the linear case in [8].

math.AP↗

arXiv2022

Reproducing sensory induced hallucinations via neural fields

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 hallucination-like stimulus. In particular, we present a theoretical framework for sensory-induced hallucinations which allows one to reproduce novel psychophysical results such as the MacKay effect (Nature, 1957) and the Billock and Tsou experiences (PNAS, 2007).

q-bio.NC↗

arXiv2022

Barrier Function-Based Adaptive Continuous Higher-Order Sliding Mode Controllers with Unbounded Perturbations

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 controllers does not require any assumptions about the perturbation; however, it can provide unbounded control gains. To ensure bounded control gains, and in the case of a Lipschitz perturbation, a second class of adaptive controllers, called the adaptive higher order Super-Twisting (HOST) algorithm, is developed.

math.OC↗

arXiv2022

Upper and lower bounds for the maximal Lyapunov exponent of singularly perturbed linear switching systems

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 tend to zero. Our approach consists in characterizing suitable auxiliary linear systems that provide lower and upper bounds for the asymptotics of the maximal Lyapunov exponent of the linear switching system as the parameter of the singular perturbation tends to zero.

math.DS↗

arXiv2022

General Nilpotent and Solvable Approximations of Almost-Riemannian Structures

It is first shown that the nilpotent or the solvable approximation of an almost-Riemannian structure at a singular point is always a linear almost-Riemannian structure on a Lie group or a homogeneous space. The generic properties of almost-Riemannian structures are then investigated in all dimensions and the generic nilpotent and solvable approximations are identified.

math.OC↗

arXiv2022

A Geometric Approach of Gradient Descent Algorithms in Linear Neural Networks

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 \emph{overfitting conjecture} which states that, for almost all training data and initial conditions, the trajectory of the corresponding gradient descent system converges to a global minimum. This would imply that the solution achieved by vanilla gradient descent algorithms is equivalent to that of the least-squares estimation, for linear neural networks of an arbitrary number of hidden layers. Built upon a key invariance property induced by the network structure, we first establish convergence of gradient descent trajectories to critical points of the square loss function in the case of linear networks of arbitrary depth. Our second result is the proof of the \emph{overfitting conjecture} in the case of single-hidden-layer linear networks with an argument based on the notion of normal hyperbolicity and under a generic property on the training data (i.e., holding for almost all training data).

cs.LG↗

arXiv2021

On the gap between deterministic and probabilistic joint spectral radii for discrete-time linear systems

Given a discrete-time linear switched system $Σ(\mathcal A)$ 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 $ρ_{\mathrm d}(\mathcal A)$ and, on the other hand, its probabilistic joint spectral radii $ρ_{\mathrm p}(ν,P,\mathcal A)$ for Markov random switching signals with transition matrix $P$ and a corresponding invariant probability $ν$. Note that $ρ_{\mathrm d}(\mathcal A)$ is larger than or equal to $ρ_{\mathrm p}(ν,P,\mathcal A)$ for every pair $(ν, P)$. In this paper, we investigate the cases of equality of $ρ_{\mathrm d}(\mathcal A)$ with either a single $ρ_{\mathrm p}(ν,P,\mathcal A)$ or with the supremum of $ρ_{\mathrm p}(ν,P,\mathcal A)$ over $(ν,P)$ and we aim at characterizing the sets $\mathcal A$ for which such equalities may occur.

math.OC↗

arXiv2021

Insights into the multiplicity-induced-dominancy for scalar delay-differential equations with two delays

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 property usually referred to as multiplicity-induced-dominancy (MID). In this paper, we investigate the MID property for one of the simplest cases of systems with two delays, a scalar delay-differential equation of first order with two delayed terms of order zero. We discuss the standard approach based on the argument principle for establishing the MID property for single-delay systems and some of its limitations in the case of our simple system with two delays, before proposing a technique based on crossing imaginary roots that allows to conclude that the MID property holds in our setting.

math.OC↗

arXiv2021

One-dimensional wave equation with set-valued boundary damping: well-posedness, asymptotic stability, and decay rates

This paper is concerned with the analysis of a one dimensional wave equation $z_{tt}-z_{xx}=0$ on $[0,1]$ with a Dirichlet condition at $x=0$ and a damping acting at $x=1$ which takes the form $(z_t(t,1),-z_x(t,1))\inΣ$ for every $t\geq 0$, where $Σ$ is a given subset of $\mathbb R^2$. The study is performed within an $L^p$ functional framework, $p\in [1, +\infty]$. We aim at determining conditions on $Σ$ ensuring existence and uniqueness of solutions of that wave equation as well as strong stability and uniform global asymptotic stability of its solutions. In the latter case, we also study the decay rates of the solutions and their optimality. We first establish a one-to-one correspondence between the solutions of that wave equation and the iterated sequences of a discrete-time dynamical system in terms of which we investigate the above mentioned issues. This enables us to provide a simple necessary and sufficient condition on $Σ$ ensuring existence and uniqueness of solutions of the wave equation as well as an efficient strategy for determining optimal decay rates when $Σ$ verifies a generalized sector condition. As an application, we solve two conjectures stated in the literature, the first one seeking a specific optimal decay rate and the second one associated with a saturation type of damping. In case the boundary damping is subject to perturbations, we derive sharp results regarding asymptotic perturbation rejection and input-to-state issues.

math.AP↗

arXiv2021

Stabilisation of the complex double integrator by means of a saturated linear feedback

Consider the saturated complex double integrator, i.e., the linear control system $\dot x=Ax+Bσ(u)$, 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 $σ:\R\to\R$ is a saturation function. We prove that there exists a linear feedback $u=K^Tx$ such that the resulting closed loop system given by $\dot x=Ax+Bσ(K^Tx)$ is globally asymptotically stable with respect to the origin.

math.OC↗