Lyapunov Stability for Linear Integral Delay Equations

Title: Simplified Lyapunov Stability Conditions for Linear Integral

Delay Equations

Laboratory: Laboratoire des signaux et syst`emes, CNRS, CentraleSup´elec, Universit´e Paris

Saclay, 91190, Gif-sur-Yvette, France

Internship Advisors: Jean AURIOL, Delphine BRESCH-PIETRI1.

Description of the Internship

I. Context and scientific objectives

Time-delay systems, which describe dynamical processes whose evolution depends on past states,

constitute a major topic in control theory. Among them, Linear Integral Delay Equations (LIDEs)

form a subclass involving both pointwise and distributed delays. Such systems naturally arise in the

modeling of engineering and biological processes subject to transport, communication, or measurement

delays (1). Relevant examples include sampled-data systems (2), population dynamics, and biomedical

applications such as epidemic models (3). Moreover, linear first-order hyperbolic Partial Differential

Equations (PDEs), widely used to represent systems governed by balance laws (4), can be reformulated

as LIDEs (5, 6).

Despite their broad applicability, LIDEs have received comparatively limited attention in the

literature, where they are often treated as a particular subclass of time-delay systems (7). Only a few

works have specifically addressed difference systems, such as (8, 9), which investigate controllability

properties and establish spectral criteria, or (10, 11, 12), which focus on deriving necessary stability

conditions.

Recently, significant progress has been achieved in this direction. Necessary Lyapunov conditions

for the exponential stability of Linear Difference Equations (LDEs) with pointwise delays were estab-

lished in (13), and Complete-Type Lyapunov functionals for LIDEs were proposed in (14), in line with

the classical framework for time-delay systems. This line of research was further extended in (15),

where an Input-to-State Lyapunov functional was introduced.

Although this functional constitutes a major theoretical advance (providing necessary and sufficient

conditions for stability), its practical applicability depends on the existence of efficient numerical

procedures for its computation, which can be highly challenging. An alternative direction is thus

to focus on sufficient conditions, allowing the design of simpler and more computationally tractable

Lyapunov functionals.

Research goal: The main objective of this internship is to derive simplified sufficient stability

conditions for LIDEs and to compare them, in terms of conservatism and computational complexity,

with the existing general Lyapunov functional framework.

II. Scientific approach

To address these questions, we will first build upon existing results for time-delay systems (2). We will

then derive simplified Lyapunov functionals based on Linear Matrix Inequalities (LMIs) and formulate

appropriate sufficient conditions guaranteeing stability. The associated numerical complexity will be

analyzed, and the conservatism of the proposed conditions will be evaluated through illustrative low-

dimensional examples.

IV. References

• (1) S.-I. Niculescu. Delay effects on stability: a robust control approach, volume 269. Springer Science & Business Media, 2001.
• (2) E. Fridman. Introduction to time-delay systems: Analysis and control. Springer, 2014.
• (3) K. Cooke and J. Kaplan. A periodicity threshold theorem for epidemics and population growth. Mathematical Biosciences, 31(1):87–104, 1976.
• (4) G. Bastin and J.-M. Coron. Stability and boundary stabilization of 1-D hyperbolic systems. Springer, 2016.
• (5) J. Auriol and F. Di Meglio. An explicit mapping from linear first order hyperbolic PDEs to difference systems. Systems & Control Letters, 123:144–150, 2019.
• (6) I. Karafyllis and M. Krstic. On the relation of delay equations to first-order hyperbolic partial differential equations. ESAIM: Control, Optimisation and Calculus of Variations, 20(3):894–923, 2014.
• (7) J.K. Hale and S.M. Verduyn Lunel. Introduction to functional differential equations. Springer-Verlag, 1993.
• (8) Y. Chitour, S. Fueyo, G. Mazanti, and M. Sigalotti. Hautus–Yamamoto criteria for approximate and exact controllability of linear difference delay equations. Discrete and Continuous Dynamical Systems, 43:3306–3337, 2023.
• (9) G. Mazanti. Relative controllability of linear difference equations. SIAM Journal on Control and Optimisation, 55(5):3132–3153, 2017.
• (10) S. Damak, M. Di Loreto, W. Lombardi, and V. Andrieu. Exponential L2-stability for a class of linear systems governed by continuous-time difference equations. Automatica, 50(12):3299–3303, 2014.
• (11) S. Damak, M. Di Loreto, and S. Mondié. Stability of linear continuous-time difference equations with distributed delay: Constructive exponential estimates. International Journal of Robust and Nonlinear Control, 25(17):3195–3209, 2015.
• (12) D. Melchor-Aguilar. Lyapunov functionals for linear continuous-time difference systems: A single delay case. Systems & Control Letters, 177: , 2023.
• (13) E. Rocha Campos, S. Mondié, and M. Di Loreto. Necessary stability conditions for linear difference equations in continuous time. IEEE Transactions on Automatic Control, 63(12):4405–4412, 2018.
• (14) R. Ortiz, A. Egorov, and S. Mondié. Robust stability analysis for integral delay systems: A complete type functional approach. IEEE Transactions on Automatic Control, 2023.
• (15) M. Lamarque, J. Auriol, and D. Bresch-Pietri. Converse Lyapunov theorem for input-to-state stability of linear integral difference equations. Automatica, 179: , 2025.

1The advisors are with Universit´e Paris-Saclay, CNRS, CentraleSup´elec, Laboratoire des signaux et syst`emes, 91190,

Gif-sur-Yvette, France and Mines Paris, PSL Research University, 75005 Paris. Emails: ,