arXiv subjects

Guilherme Mazanti

Publications and source records attributed to Guilherme Mazanti.

At least 19 recordsLinked to original sources

arXiv2026

Strong Stability of Linear Functional Equations with Distributed Delays

This paper considers linear functional equations on $\mathbb R^d$ with distributed delays defined by matrix-valued measures of bounded variation. More precisely, we are interested in providing conditions to ensure that the exponential stability of these systems is preserved under small changes of the parameters which define them. In the special case of difference equations, it is known that exponential stability is preserved under small perturbations of the matrices defining the system, but not of the delays, and an additional condition for preservation of exponential stability under perturbation of the delays is given by the Hale--Silkowski criterion (HSC). In this paper, we extend the treatment of these issues to more general systems. For that purpose, we first put forward an appropriate definition of perturbation on the delays and then propose a conjecture in the spirit of (HSC). We prove several partial results related to that conjecture.

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arXiv2026

A variational mean field game of controls with free final time and pairwise interactions

This article considers a mean field game model inspired by crowd motion models in which agents aim at reaching a given target set and wish to minimize a cost consisting of an individual running cost, an individual cost depending on the arrival time at the target set, and an interaction running cost, which takes the form of pairwise interactions with other agents through both positions and velocities. We subsume this game under a more general class of games on abstract Polish spaces with pairwise interactions, and prove that the latter games have a variational structure (in the sense that their equilibria can be characterized as critical points of some potential functional) and admit equilibria. We also discuss two a priori distinct notions of equilibria, providing a sufficient condition under which both notions coincide. The results for the games in abstract Polish spaces are applied to our mean field game model, and a numerical illustration concludes the paper.

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arXiv2026

From Nash to Cournot--Nash equilibria via $Γ$-convergence

This work addresses the issue of the convergence of an $N$-player game towards a limit model involving a continuum of players, as the number of agents $N$ goes to infinity. More precisely, we investigate the convergence of Nash equilibria to a Cournot--Nash equilibrium of the limit model. When the cost function of the players is the first variation of some potential function, equilibria can be characterized by a stationarity condition, satisfied in particular by the minimizers of the potential. We demonstrate such a characterization under low regularity assumptions. Then we focus on the case where the players interact in a pairwise fashion; in this case we show that the original sequence of $N$-player games also admit a potential structure and prove that their corresponding potential functions converge in the sense of $Γ$-convergence to the potential function of the limit game.

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arXiv2026

A mathematical framework for time-delay reservoir computing analysis

Reservoir computing is a well-established approach for processing data with a much lower complexity compared to traditional neural networks. Despite two decades of experimental progress, the core properties of reservoir computing (namely separation, robustness, and fading memory) still lack rigorous mathematical foundations. This paper addresses this gap by providing a control-theoretic framework for the analysis of time-delay-based reservoir computers. We introduce formal definitions of the separation property and fading memory in terms of functional norms, and establish their connection to well-known stability notions for time-delay systems as incremental input-to-state stability. For a class of linear reservoirs, we derive an explicit lower bound for the separation distance via Fourier analysis, offering a computable criterion for reservoir design. Numerical results on the NARMA10 benchmark and continuous-time system prediction validate the approach with a minimal digital implementation.

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arXiv2025

Hautus--Yamamoto criteria for approximate and exact controllability of linear difference delay equations

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 $t-Λ_1,\dots,t-Λ_N$. Based on the realization theory developed by Y.Yamamoto for general infinite-dimensional dynamical systems, we obtain necessary and sufficient conditions, expressed in the frequency domain, for the approximate controllability in finite time in $L^q$ spaces, $q \in [1, +\infty)$. We also provide a necessary condition for $L^1$ exact controllability, which can be seen as the closure of the $L^1$ approximate controllability criterion. Furthermore, we provide an explicit upper bound on the minimal times of approximate and exact controllability, given by $d\max\{Λ_1,\dots,Λ_N\}$, where $d$ is the dimension of the state space.

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arXiv2025

Strong stability of linear delay-difference equations

This paper considers linear delay-difference equations, that is, equations relating the state at a given time with its past values over a given bounded interval. After providing a well-posedness result and recalling Hale--Silkowski Criterion for strong stability in the case of equations with finitely many pointwise delays, we propose a generalization of the notion of strong stability to the more general class of linear delay-difference equations with an integral term defined by a matrix-valued measure. Our main result is an extension of Melvin Criterion for the strong stability of scalar equations, showing that local and global strong stability are equivalent, and that they can be characterized in terms of the total variation of the function defining the equation. We also provide numerical illustrations of our main result.

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arXiv2025

Well-posedness and asymptotic behavior of difference equations with a time-dependent delay and applications

In this paper, we investigate the well-posedness and asymptotic behavior of difference equations of the form $x(t) = A x(t - τ(t))$, $t \geq 0$, where the unknown function $x$ takes values in $\mathbb R^d$ for some positive integer $d$, $A$ is a $d \times d$ matrix with real coefficients, and $τ\colon [0, +\infty) \to (0, +\infty)$ is a time-dependent delay. We provide our investigations for three spaces of functions: continuous, regulated, and $L^p$. We compare our results for these three cases, showing how the hypotheses change according to the space that we are treating. Finally, we provide applications of our results to difference equations with state-dependent delays for the cases of continuous and regulated function spaces, as well as to transport equations in one space dimension with time-dependent velocity.

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arXiv2025

A novel necessary and sufficient condition for the stability of $2\times 2$ first-order linear hyperbolic systems

In this paper, we establish a necessary and sufficient stability condition for a class of two coupled first-order linear hyperbolic partial differential equations. Through a backstepping transform, the problem is reformulated as a stability problem for an integral difference equation, that is, a difference equation with distributed delay. Building upon a Stépán--Hassard argument variation theorem originally designed for time-delay systems of retarded type, we then introduce a theorem that counts the number of unstable roots of our integral difference equation. This leads to the expected necessary and sufficient stability criterion for the system of first-order linear hyperbolic partial differential equations. Finally, we validate our theoretical findings through simulations.

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arXiv2024

A note on existence and asymptotic behavior of Lagrangian equilibria for first-order optimal-exit mean field games

In this paper, we consider a first-order mean field game model motivated by crowd motion in which agents evolve in a (not necessarily compact) metric space and wish to reach a given target set. Each agent aims to minimize the sum of their travel time and an exit cost which depends on their exit position on the target set. Agents interact through their dynamics, the maximal speed of an agent being assumed to be a function of their position and the distribution of other agents. This interaction may model, in particular, congestion phenomena. Under suitable assumptions on the model, we prove existence of Lagrangian equilibria, analyze the asymptotic behavior for large time of the distribution of agents, and study the dependence of equilibria and asymptotic limits on the initial distribution of the agents.

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arXiv2024

Approximate and exact controllability criteria for linear one-dimensional hyperbolic systems

This paper deals with the controllability of linear one-dimensional hyperbolic systems. Reformulating the problem in terms of linear difference equations and making use of infinite-dimensional realization theory, we obtain both necessary and sufficient conditions for approximate and exact controllability, expressed in the frequency domain. The results are applied to flows in networks.

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arXiv2024

A nonsmooth Frank-Wolfe algorithm through a dual cutting-plane approach

An extension of the Frank-Wolfe Algorithm (FWA), also known as Conditional Gradient algorithm, is proposed. In its standard form, the FWA allows to solve constrained optimization problems involving $β$-smooth cost functions, calling at each iteration a Linear Minimization Oracle. More specifically, the oracle solves a problem obtained by linearization of the original cost function. The algorithm designed and investigated in this article, named Dualized Level-Set (DLS) algorithm, extends the FWA and allows to address a class of nonsmooth costs, involving in particular support functions. The key idea behind the construction of the DLS method is a general interpretation of the FWA as a cutting-plane algorithm, from the dual point of view. The DLS algorithm essentially results from a dualization of a specific cutting-plane algorithm, based on projections on some level sets. The DLS algorithm generates a sequence of primal-dual candidates, and we prove that the corresponding primal-dual gap converges with a rate of $O(1/\sqrt{t})$.

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arXiv2023

Frequency domain approach for the stability analysis of a fast hyperbolic PDE coupled with a slow ODE

This paper deals with the exponential stability of systems made of a hyperbolic PDE coupled with an ODE with different time scales, the dynamics of the PDE being much faster than that of the ODE. Such a difference of time scales is modeled though a small parameter $\varepsilon$ multiplying the time derivative in the PDE, and our stability analysis relies on the singular perturbation method. More precisely, we define two subsystems: a reduced order system, representing the dynamics of the full system in the limit $\varepsilon = 0$, and a boundary-layer system, which represents the dynamics of the PDE in the fast time scale. Our main result shows that, if both the reduced order and the boundary-layer systems are exponentially stable, then the full system is also exponentially stable for $\varepsilon$ small enough, and our strategy is based on a spectral analysis of the systems under consideration. Our main result improves a previous result in the literature, which was proved using a Lyapunov approach and required a stronger assumption on the boundary-layer system to obtain the same conclusion.

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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.

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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.

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arXiv2022

MID Property for Delay Systems: Insights on Spectral Values with Intermediate Multiplicity

This paper focuses on the problem of multiplicity induced dominancy (MID) for a class of linear time-invariant systems represented by delay-differential equations. If the problem of generic MID was characterized in terms of properties of the roots of Kummer hypergeometric functions, the case of intermediate MID is still an open problem. The aim of this paper is to address such a problem by using the Green--Hille transformation for characterizing the distribution of the nonasymptotic zeros of linear combinations of Kummer functions. An illustrative example completes the presentation and shows the effectiveness of the proposed methodology.

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arXiv2022

YALTAPy and YALTAPy_Online: Python toolboxes for the $H_\infty$-stability analysis of classical and fractional systems with commensurate delays

The aim of this paper is to give a presentation of the Python toolbox YALTAPy dedicated to the stability study of standard and fractional delay systems as well as its online version YALTAPy_Online. Both toolboxes are derived from YALTA whose functionalities will be recalled here. Examples will be given to show how these toolboxes may be used.

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arXiv2022

Padé Approximation and Hypergeometric Functions: A Missing Link with the Spectrum of Delay-Differential Equations

It is well known that rational approximation theory involves degenerate hypergeometric functions and, in particular, the Padé approximation of the exponential function is closely related to Kummer hypergeometric functions. Recently, in the context of the study of the exponential stability of the trivial solution of delay-differential equations, a new link between the degenerate hypergeometric function and the zeros distribution of the characteristic function associated with linear delay-differential equations was emphasized. Such a link allowed the characterization of a property of time-delay systems known as multiplicity-induced-dominancy (MID), which opened a new direction in designing low-complexity controllers for time-delay systems by using a partial pole placement idea. Thanks to their relations to hypergeometric functions, we explore in this paper links between the spectrum of delay-differential equations and Padé approximations of the exponential function. This note exploits and further comments recent results from [I. Boussaada, G. Mazanti and S-I. Niculescu. 2022, Comptes Rendus. Mathématique, arXiv:2107.11363] and [I. Boussaada, G. Mazanti and S-I. Niculescu. 2022, Bulletin des Sciences Mathématiques, arXiv:2106.03378].

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arXiv2022

New Features of P3$δ$ Software. Insights and Demos

This paper presents the software entitled "Partial Pole Placement via Delay Action", or "P3$δ$" for short. P3$δ$ is a Python software with a friendly user interface for the design of parametric stabilizing feedback laws with time-delays for dynamical systems. After recalling the theoretical foundation of the so-called "Partial Pole Placement" methodology we propose as well the main features of the current version of P3$δ$. We illustrate its use in feedback stabilization of several control systems operating under time delays.

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