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.
III. Application
To apply, write an email with your CV and a transcript to Jean Auriol, Delphine Bresch-Pietri.
IV. References
[1] S.-I. Niculescu. Delay effects on stability: a robust control approach, volume 269. Springer Science &
Business Media, 2001.
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: jean.auriol@centralesupelec.fr,
delphine.bresch-pietri@minesparis.psl.eu.
[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. Mathe-
matical 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 Opti-
mization, 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. Mondi´e. 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, 177:105536, 2023.
[13] E. Rocha Campos, S. Mondi´e, 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. Mondi´e. 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, 179:112437, 2025.
