2025Activity​​ reportProject-TeamDISCO

2.1‌ Objectives

Control of interacting‌​‌ complex natural or artificial​​ systems guaranteeing performance, safety​​​‌ and low computational burden‌ is a challenge for‌​‌ the years to come.​​

Things have drastically changed​​​‌ from a material point‌ of view in the‌​‌ last 15 years in​​ engineering, biological and medical​​​‌ fields. For instance, in‌ the field of robotics,‌​‌ the time when large​​ robots, evolving in secure​​​‌ spaces where no human‌ was supposed to be‌​‌ able to enter, could​​ be managed with very​​​‌ basic control laws is‌ over. Developing control algorithms‌​‌ for robots of smaller​​ size and having to​​​‌ adapt online their behaviour‌ to the dynamics of‌​‌ the environment is much​​ more difficult.

These topics​​​‌ are clearly the same‌ in fields such as‌​‌ autonomous vehicles, unmanned aerial​​ vehicles, traffic and energy​​​‌ networks etc. Moreover, in‌ the case where several‌​‌ robots/vehicles/agents are cooperating, the​​ question of communication is​​​‌ crucial and delays as‌ well as disconnection are‌​‌ major issues in this​​ context. The size of​​​‌ the network is certainly‌ another key question.

Adopting‌​‌ a model-based approach, the​​ aim of the team​​​‌ is to develop (optimal)‌ control methods for (possibly‌​‌ large size) interconnected systems​​ with the ultimate goal​​​‌ to produce implementable and‌ low computational burden solutions‌​‌ (in the modest sense​​ of favoring low complexity​​​‌ controllers from an implementation‌ point of view).‌​‌

We contribute to the​​ modeling of chosen applications​​​‌ in Energy, bio-systems, Engineering‌ and medicine. Partial Differential‌​‌ Equations (PDEs), Ordinary Differential​​ Equations (ODEs) (possibly with​​​‌ delays), models will be‌ considered in the linear‌​‌ as well as the​​ nonlinear setting. Discretization of​​​‌ PDEs as well as‌ the modeling of discrete‌​‌ measurements/actuation arising in continuous​​ model systems will give​​​‌ rise to infinite or‌ finite-dimensional discrete-time models. We‌​‌ also consider general classes​​ of systems which encompass​​​‌ the framework of our‌ applications such as Fractional‌​‌ Differential or Difference Equations.​​

Our goal is to​​​‌ analyze these models as‌ much as possible without‌​‌ further simplifications in order​​ to capture most of​​​‌ the phenomena (and, in‌ particular, time-heterogeneity) and to‌​‌ develop control algorithms for​​ them. This includes stability​​​‌ analysis, observation, robustness analysis‌ and (optimal) control.‌​‌

Note that we also​​ perform some research at​​​‌ the confluence of Control‌ Methods and Machine Learning‌​‌ (ML), not developing ourselves​​ ML techniques to synthesize​​​‌ controllers from data but‌ rather studying how System‌​‌ Theory and Optimization Theory​​ can help analyzing closed-loop​​​‌ systems containing Neural Networks-based‌ controllers.

3.1 Stability Analysis and​​ Control of Linear Interconnected​​​‌ Systems

This first axis‌ is devoted to the‌​‌ study of stability properties​​​‌ of (a small number​ of) interconnected linear systems,​‌ an area where many​​ fundamental questions are still​​​‌ open. Algebraic tools, time-domain​ (including semi-group Theory) and​‌ frequency domain (where Hypergeometric​​ functions should play an​​​‌ important role) techniques are​ involved to characterize stabilizability​‌ and controllability properties as​​ well as asymptotic, exponential,​​​‌ $H_{\infty }$ and ${\mathrm{L}}^p$ (in particular ${\mathrm{L‌}}^{\infty }$i.e. BIBO)-stability properties.​​

For standard or fractional​​​‌ systems with delays,​ we are particularly interested​‌ in questions such as​​ :

• The location in​​​‌ the complex plane of​ the roots of quasi-polynomials​‌ of the type ${\displaystyle {\mathrm{p}}_0(s)‌+\sum _{i=1}^n{p}_{\mathrm{i‌}}\left(s\right){\mathrm{e}}^{-h_i\mathrm{s‌}}}$ where $p_i\mathrm{i}=0,\cdots ‌,n$ are real​​ polynomials in the complex​​​‌ variable $s$ satisfying $degp_0\ge deg‌p_i$ and ${\mathrm{h}}_i>0$.​​​‌
• The stability properties of​ linear delay systems with​‌ time-varying delays and/or time​​ varying coefficients via both​​​‌ time-domain and input-output methods.​ Robustness issues will be​‌ investigated.
• Structural properties of​​ time-delay systems, such as​​​‌ controllability or stabilizability.
• Partial​ pole placement for delay​‌ systems : we aim​​ here at establishing a​​​‌ general approach to control​ the decay rate of​‌ the solutions via the​​ assignment of the spectral​​​‌ abscissa.

For Systems Modeled​ by Partial Differential Equations​‌, we want to​​ perform the analysis of​​​‌ :

• Stability properties of​ propagation phenomena modeled by​‌ partial differential equations.
• Partial​​ differential equations with different​​​‌ time scales
• Dynamical systems​ with Integral delay equations​‌
• Delayed control of systems​​ modeled by PDEs

3.2​​​‌ Observation and Control of​ Nonlinear Interconnected Systems

The​‌ second axis explores the​​ control and observation (in​​​‌ particular finite-time observation) of​ various classes of (a​‌ small number of interconnected)​​ nonlinear delay systems. Lyapunov​​​‌ analysis will be a​ central tool.

Concerning the​‌ analysis of nonlinear systems​​, we will focus​​​‌ on:

• The local analysis​ of nonlinear systems with​‌ delays in particular via​​ the trajectory-based approach.

• The​​​‌ stability analysis and performance​ assessment of Piecewise Affine​‌ systems described by the​​ equations

  $$\begin{array}{ccc}\hfill x[\mathrm{k‌}+1]& =& F_{1}x[‌k]+{\mathrm{F}}_2\varphi \left(\mathrm{y‌}\left(x\left[\mathrm{k}\right]\right)\right)\hfill \\ \hfill \mathrm{y‌}\left(x\right[\mathrm{k}\left]\right)& =& {\mathrm{F‌}}_{3}x\left[\mathrm{k}\right]+F_{\mathrm{4‌}}\varphi \left(y(x[k]‌)\right)+{\mathrm{f}}_5\hfill \end{array}$$

  where the vector-valued​‌ function $\varphi :{\mathrm{ℝ}}^m\to {ℝ}^{\mathrm{m‌}}$ is a vector of​ ramp functions on each​‌ argument.

• Dynamical systems with​​ Neural Networks (NN) in​​​‌ the loop

Concerning observers​ and estimation of parameters​‌, we wish to​​ develop:

• Finite-time observers and​​​‌ estimation of unknown parameters​ for classes of systems​‌ that encompass discrete-time systems​​ and systems with unknown​​​‌ parameters in the measurements.​
• Advanced estimation and control​‌ techniques for traffic networks​​ (new mesoscopic models will​​​‌ be developed).

3.3 Optimal​ control of mean-field type​‌ dynamical systems

Our research​​ activity in optimal control​​ is motivated by large-scale​​​‌ problems and aims at‌ addressing both theoretical and‌​‌ numerical aspects. We have​​ a focus on situations​​​‌ involving a large number‌ of interacting entities: for‌​‌ example, a fleet of​​ electrical vehicles, a cell​​​‌ population in biology, or‌ a large training set‌​‌ in supervized learning. In​​ such situations, the mean-field​​​‌ approximation consists in considering‌ the probability distribution of‌​‌ these entities rather than​​ an enumeration of their​​​‌ states. This point of‌ view allows for a‌​‌ simpler mathematical treatment and​​ numerical resolution.

In the​​​‌ Mean Field Game domain‌, we want to:‌​‌

• Address the case of​​ Agents with constraints and​​​‌ free-final time
• Study a‌ new class of interactions‌​‌ for MFGs, which we​​ called pairwise interactions the​​​‌ cost of each agent‌ is the sum, with‌​‌ respect to all other​​ agents, of some interaction​​​‌ function of the considered‌ agent and any other‌​‌ agent.
• Develop variants of​​ the Frank-Wolfe algorithm (whose​​​‌ interest go beyond MFGs)‌ which are capable to‌​‌ address constrained or non-smooth​​ MFGs.
• Export of our​​​‌ expertise on classical MFG‌ models to more applied‌​‌ problems, in interactions with​​ other mathematical fields :​​​‌ Management of fisheries, Energy,‌ biology

Concerning the Theoretical‌​‌ analysis of some modern​​ machine-learning methods, we​​​‌ aim to to continue‌ to exploit their natural‌​‌ connection to Mean Field​​ Models by analyzing:

• Some​​​‌ machine learning methods that‌ explicitly incorporate terms that‌​‌ can be interpreted as​​ mean-field interactions.
• Deep neural​​​‌ networks that may be‌ viewed as the discretization‌​‌ of ODE’s
• Neural networks​​ that may be viewed​​​‌ as the discretization of‌ PDEs (such as convolutional‌​‌ residual networks)
• Transformers, seens​​ as a function on​​​‌ the set of probability‌ measures.

4.1 Analysis and Control​​ of life sciences systems​​​‌

The team is involved‌ in life sciences applications.‌​‌ The actual two main​​ lines are the analysis​​​‌ of bioreactors models (microorganisms;‌ bacteria, microalgae, yeast, etc..)‌​‌ and the management of​​ fisheries.

4.2 Energy Systems​​​‌

4.3 Mechanical‌ systems

4.4 Transportation Systems​​​‌

The team is interested​ in control applications in​‌ transportation systems. In particular,​​ the problem of collision​​​‌ avoidance of autonomous vehicels​ has been investigated under​‌ the framework of Time​​ Varying systems. The goal​​​‌ is to obtain closed-loop​ control laws that guarantee​‌ the execution of a​​ trajectory under uncertainties such​​​‌ as road and vehicle​ conditions.

5.1 Awards​​

Silviu-Iulian Niculescu has been​​​‌ elected fellow of IFAC​ (International Federation of Automatic​‌ Control) to acknowledge his​​ outstanding contributions in the​​​‌ fields of interest of​ IFAC (namely, for contributions​‌ to the analysis and​​ control of dynamical systems​​​‌ with delays).

6.1 Stability​‌ of delay systems with​​ delay-dependent coefficients

Participants: Islam​​​‌ Boussaada, Silviu Niculescu​, Keqin Gu [Illinois]​‌, Jin Chi [TuSimple​​ Inc.], Quian Ma​​​‌ [Nanjing].

We developed​ in 13 a method​‌ of stability analysis of​​ linear time-delay systems with​​​‌ commensurate delays and delay-dependent​ coefficients. The method is​‌ based on a D-decomposition​​ formulation that consists of​​​‌ identifying the critical pairs​ of delay and frequency,​‌ and determining the corresponding​​ crossing directions. The process​​​‌ of identifying the critical​ pairs consists of a​‌ magnitude condition and a​​ phase condition. The magnitude​​​‌ condition utilizes the Orlando’s​ formula, and generates frequency​‌ curves within the delay​​ interval of interest. Such​​​‌ frequency curves correspond to​ the delay-frequency pairs such​‌ that the decomposition equation​​ has at least one​​​‌ solution on the unit​ circle. The delay interval​‌ of interest is divided​​ into continuous frequency curve​​​‌ intervals (CFCIs). Under some​ non-degeneracy assumptions, the number​‌ of frequency curves remains​​ constant within each CFCI,​​​‌ and the associated decomposition​ equation has one and​‌ only one solution on​​ the unit circle at​​​‌ any point on a​ frequency curve. By traversing​‌ through the frequency curves,​​ all the crossing points​​​‌ can be identified. The​ crossing direction is related​‌ to the sign of​​ the lowest-order nonzero derivative​​​‌ of the phase angle​ with respect to the​‌ delay, which is a​​ generalization of the existing​​​‌ literature even for the​ case with single delay.​‌ This conclusion allows one​​ to determine the crossing​​​‌ direction by examining the​ phase angle vs delay​‌ diagram.

6.2 Partial poles​​ placement and observer design​​​‌ in delay systems

Participants:​ Islam Boussaada, Silviu​‌ Niculescu, Ahlem Sassi​​ [ESME], Michel Zasadzinski​​​‌ [Nancy].

We extended​ in 37, 45​‌ the use of the​​ pole placement ap- proach​​​‌ in the design of​ observers for certain classes​‌ of Linear-Time-Invariant (LTI) systems​​ with time delay. Specifically,​​​‌ depending on how the​ delay appears in the​‌ system dynamics, two classes​​ of dynamic systems are​​​‌ considered: state-delayed systems and​ input-delayed systems, respectively. First,​‌ we address the problem​​ of designing a full-order​​ Luenberger observer for the​​​‌ considered systems using partial‌ placement of the error‌​‌ poles. Namely, we exploit​​ the multiplicity-induced-dominancy (MID) property​​​‌ of the characteristic root‌ with the maximal admissible‌​‌ nultiplicity of the characteristic​​ function corresponding to the​​​‌ system’s error. After giving‌ the existing condition for‌​‌ the proposed observer, we​​ use the same MID​​​‌ property but in the‌ so-called generic case. The‌​‌ performance and effectiveness of​​ the proposed observers are​​​‌ highlighted through several illustrative‌ examples.

6.3 Robustness filters‌​‌ for the Smith Predictor​​

Participants: Giorgio Valmorbida,​​​‌ Daniel Amaral, Bismarck‌ Torrico [Universidade Federal do‌​‌ Ceará], Mattia Mattioni​​ [Universidade Federal do Ceará]​​​‌.

We have studied‌ the discrete-time version of‌​‌ the Smith predictor with​​ the goal of designing​​​‌ a robustness filter. The‌ goals for robustness are‌​‌ to guarantee disturbance rejection​​ levels in presence of​​​‌ parametric uncertainties. The proposed‌ methods for filter design‌​‌ guarantee the disturbance minimization.​​

6.4 Stabilization of time-varying​​​‌ delay systems

Participants: Frédéric‌ Mazenc, Catherine Bonnet‌​‌.

In the contribution​​ 30, we proposed​​​‌ two control designs of‌ exponentially stabilizing feedbacks for‌​‌ two families of time-varying​​ continuous-time systems with a​​​‌ pointwise delay in the‌ input. A smallness condition‌​‌ is imposed on the​​ delay. In contrast to​​​‌ the classical reduction model‌ approach, the results do‌​‌ not necessitate knowledge of​​ the state transition matrix,​​​‌ which is crucial because,‌ in general, no explicit‌​‌ formula for state transition​​ matrices can be determined.​​​‌ The price to pay‌ for this is a‌​‌ limitation on the size​​ of the delay.

6.5​​​‌ Interplay between discretization and‌ controllability of linear delay‌​‌ systems: An algebraic viewpoint​​

Participants: Silviu-Iulian Niculescu,​​​‌ Hugues Mounier [L2S],‌ Florentina Nicolau [ENSEA Cergy]‌​‌.

In the paper​​ 21, the authors​​​‌ give an in depth‌ study of linear delay‌​‌ systems controllability preservation/alteration through​​ discretization. We make use​​​‌ of a module theoretic‌ framework acting as a‌​‌ uni- fying one for​​ most of the existing​​​‌ delay system controllability notions.‌ We propose a formal‌​‌ generic definition of a​​ discretization scheme and illustrate​​​‌ through examples that controllability‌ properties may be lost‌​‌ through discretization. Then, we​​ introduce the notion of​​​‌ preservation (that is, a‌ measure of quantifying the‌​‌ ability of the discretizer​​ to preserve control- lability​​​‌ properties) and prove that‌ for a given discretizer,‌​‌ we can always find​​ a delay system for​​​‌ which even the torsion-free‌ controllability (which is the‌​‌ weakest controllability notion) is​​ not preserved. Finally, we​​​‌ reverse the situation, and‌ show that for any‌​‌ given delay system, preserving​​ discretizers exist.

6.6 Strong​​​‌ stability of linear delay-difference‌ equations

Participants: Yacine Chitour‌​‌ [L2S], Felipe Gonçalves​​ Netto, Guilherme Mazanti​​​‌.

Linear delay-difference equations‌ have been previously considered‌​‌ in the literature due​​ to the facts that,​​​‌ on the one hand,‌ the analysis of some‌​‌ hyperbolic partial differential equations​​ (PDEs) can be reduced​​​‌ to this class of‌ systems and, on the‌​‌ other hand, their stability​​ analysis also provides information​​​‌ on more general time-delay‌ systems of neutral type‌​‌ 62, 64,​​​‌ 60, 61,​ 59, 53.​‌

An importat observation about​​ neutral time-delay systems and​​​‌ delay-difference equations is that​ some of their properties,​‌ such as exponential stability,​​ are not robust with​​​‌ respect to arbitrarily small​ perturbations on some of​‌ the parameters of the​​ system, namely the delays​​​‌ 62, 68.​ This has motivated the​‌ definition of stronger notions​​ of stability, such as​​​‌ strong stability 62.​ In the literature, strong​‌ stability had been previously​​ considered only for the​​​‌ case of perturbations of​ pointwise delays.

In 12​‌, 49, we​​ have first identified a​​​‌ suitable notion of perturbation​ of the delays that​‌ also extends to distributed​​ delays, extending thus the​​​‌ definition of strong stability​ to a wider class​‌ of delay-difference equations. We​​ have then proved in​​​‌ 12 that the classical​ Hale–Silkowski strong stability criterion​‌ (see, e.g., 54,​​ 62, 68)​​​‌ can be extended to​ this new setting, at​‌ least for scalar equations.​​ In the general setting​​​‌ of systems of equations,​ many partial results are​‌ provided in 49,​​ proving that the Hale–Silkowski​​​‌ criterion also holds when​ the definition of strong​‌ stability is slightly modified​​ in order to include​​​‌ a uniform rate of​ convergence, and up to​‌ considering only systems defined​​ by measures with no​​​‌ singular part.

6.7 Delay-difference​ equations with varying delays​‌

Participants: Guilherme Mazanti,​​ Jaqueline Godoy Mesquita [UNICAMP,​​​‌ Brazil].

While the​ stability analysis of delay-differential​‌ equations with constant delays​​ has been studied from​​​‌ several perspectives in the​ literature 62, 54​‌, 60, 63​​, 67, 64​​​‌, many subtleties appear​ when considering time- or​‌ state-dependent delays, as already​​ pointed out in 55​​​‌.

The work 52​ provides a detailed analysis​‌ of the simplest form​​ of a linear delay-difference​​​‌ equation with a single​ time-varying delay, namely $\mathrm{x‌}\left(t\right)=Ax(\mathrm{t‌}-\tau (\mathrm{t}\left)\right)$, where​‌ $x(t)\in {ℝ}^d$ is​​​‌ the instantaneous state at​ time $t$, $\mathrm{A‌}$ is a given $\mathrm{d}\times d$ matrix with​​​‌ real coefficients, and $\mathrm{\tau }(·)$ is​‌ a time-varying delay. The​​ main goal of the​​​‌ work is to study​ the well-posedness of such​‌ a system and to​​ characterize the asymptotic behavior​​​‌ of its solutions. We​ provide our investigations for​‌ three types of functional​​ spaces: continuous functions, regulated​​​‌ functions, and $L^{\mathrm{p}}$ functions, $p\in [‌1,+\mathrm{\infty }]$. We derive​​​‌ sufficient conditions for well-posedness​ that are, in most​‌ cases, close to being​​ necessary, and our detailed​​​‌ analysis illustrates interesting phenomena​ specific to time-varying delays,​‌ such as the fact​​ that an equation can​​​‌ be exponentially stable in​ some $L^p$ space​‌ but unstable in another​​ $L^q$ space, $\mathrm{q‌}\ne p$. 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.

6.8 Prescribed‌ exponential stabilization of infinite‌​‌ dimensional systems

Participants: Islam​​ Boussaada, Guilherme Mazanti​​​‌, Silviu-Iulian Niculescu,‌ Timothée Schmoderer, Kaïs‌​‌ Ammari [Monastir], Fazia​​ Bedouhene [Tizi-Ouzou], Wim​​​‌ Michiels [Leuven], Cyprien‌ Tamekue [IPSA], Sami‌​‌ Tliba [UPSaclay], Karim​​ Trabelsi [IPSA].

The​​​‌ recently developed prescribed stabilization‌ paradigm, which has been‌​‌ successfully applied to the​​ control of certain classes​​​‌ of hyperbolic partial differential‌ equations 3, 2‌​‌, originates from a​​ spectral technique known as​​​‌ partial poles placement (PPP).‌ This method was initially‌​‌ introduced in the context​​ of linear time-invariant functional​​​‌ differential equations; see, for‌ instance, 58. Unlike‌​‌ standard pole placement for​​ finite-dimensional systems, which reduces​​​‌ to a classical interpolation‌ problem, PPP is considerably‌​‌ more subtle 57.​​ In particular, PPP relies​​​‌ on a remarkable spectral‌ property of delay equations,‌​‌ referred to as multiplicity-induced-dominancy​​ (MID) and coexistent-real-roots-induced-dominancy (CRRID).​​​‌ In particular, in the‌ context of the MID‌​‌ propoerty, when the multiplicity​​ of a given spectral​​​‌ value coincides with the‌ degree of the characteristic‌​‌ quasipolynomial, this property is​​ called generic MID (GMID),​​​‌ in opposition to the‌ intermediate MID (IMID), which‌​‌ corresponds to a multiplicity​​ strictly smaller than the​​​‌ degree.

Motivated by recently‌ obtained factorizations of characteristic‌​‌ functions satisfying the GMID​​ property in terms of​​​‌ Kummer confluent hypergeometric functions,‌ we provided in 8‌​‌ a new representation of​​ some Kummer functions with​​​‌ integer coefficients in terms‌ of iterated integrals of‌​‌ an exponential kernel. As​​ a consequence of the​​​‌ existing links between Kummer,‌ Whittaker, and modified Bessel‌​‌ functions, the latter classes​​ of special functions also​​​‌ take advantage of such‌ an iterated integral representation.‌​‌ We also express characteristic​​ functions of the aforementioned​​​‌ classes of time-delay systems‌ in terms of iterated‌​‌ integrals, and illustrate how​​ such an iterated integral​​​‌ representation allows us to‌ obtain information on the‌​‌ location of the spectrum​​ of the system, at​​​‌ least in some low-order‌ cases.

Furthermore, it is‌​‌ shown in 19 that​​ the multiple root configuration​​​‌ corresponds not only to‌ rightmost roots, but also‌​‌ to global minimizers of​​ the spectrum abscissa function.​​​‌ A computational characterization of‌ minima of the spectral‌​‌ abscissa is made for​​ output feedback, yielding a​​​‌ more complex picture, which‌ includes configurations with both‌​‌ multiple and simple rightmost​​ roots. In the analysis,​​​‌ the pivotal role of‌ the invariant zeros is‌​‌ highlighted, which translates into​​ restrictions on the tunable​​​‌ parameters in the closed-loop‌ quasi-polynomial.

In 9 the‌​‌ IMID property is investigated​​ for spectral values with​​​‌ the lowest overorder intermediate‌ multiplicity, i.e., a multiplicity‌​‌ larger than the order​​ of the DDE. We​​​‌ highlight the fact that‌ a root of overorder‌​‌ multiplicity is necessarily a​​ root of a particular​​​‌ polynomial, called the elimination-produced‌ polynomial (EPP), and we‌​‌ address the MID property​​ using a suitable factorization​​​‌ of the corresponding characteristic‌ function involving special functions‌​‌ of Kummer type. Additional​​ results and discussion are​​​‌ provided in the case‌ of the nth order‌​‌ integrator, in particular on​​​‌ the local optimality of​ a multiple root. The​‌ derived results show how​​ the delay can be​​​‌ further exploited as a​ control parameter and are​‌ applied to some problems​​ of stabilization of standard​​​‌ benchmarks with prescribed exponential​ decay. Next, by exploiting​‌ the concept of the​​ EPP in 10,​​​‌ the IMID property is​ fully characterized thanks to​‌ the representation of the​​ corresponding characteristic function as​​​‌ a linear combinations of​ Kummer functions.

The MID​‌ property is investigated in​​ the multidimensional context in​​​‌ 7, 42,​ where its validity is​‌ characterized in the planar​​ case, then exploited in​​​‌ the design of a​ state-feedback achieving a prescribed​‌ exponential stabilization of the​​ solution of planar dynamical​​​‌ systems with input delay.​

Finally, in the context​‌ of partial differential equations,​​ the prescribed stabilization is​​​‌ applied in 1 in​ the stabilization of the​‌ wave equation with pointwise​​ delay feedback with the​​​‌ advantage of the assignation​ of the closed-loop exponential​‌ decay. The methodology involves​​ a four-parameter autoregressive control​​​‌ structure for which the​ design strategy is based​‌ on multiplicity manifolds. An​​ extension of the MID​​​‌ property for continuous-time difference​ equations is obtained.

6.9​‌ Stability and stabilization of​​ partial differential equations

Participants:​​​‌ Islam Boussaada, Guilherme​ Mazanti, Ismaila Balogoun​‌, Jean Auriol [L2S]​​.

We establish in​​​‌ 5 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.​‌

6.10 Extremum seeking

Participants:​​ Frédéric Mazenc, Michael​​​‌ Malisoff [LSU], Emilia​ Fridman [Tel Aviv Univ]​‌.

In the contribution​​ 18, we developed​​​‌ a novel stability analysis​ for a bounded-gradient-based extremum​‌ seeking scheme applied to​​ two-variable static quadratic maps​​​‌ in the presence of​ time-varying additive measurement uncertainty.​‌ Unlike earlier approaches that​​ rely on averaging techniques,​​​‌ our method is based​ on a new state​‌ transformation combined with a​​ time-varying quadratic Lyapunov function​​​‌ and a comparison principle.​ This framework allowed us​‌ to derive significantly less​​ conservative bounds on both​​​‌ the dither frequency and​ the ultimate bound of​‌ the estimation error than​​ those reported in previous​​​‌ contributions. The effectiveness and​ reduced conservatism of the​‌ proposed approach were numerically​​ validated through a representative​​​‌ example.

6.11 Construction of​ strict Lyapunov functions

Participants:​‌ Frédéric Mazenc, Mohamed​​ Maghenem [GIPSA-Lab], Antonio​​​‌ Loria [L2S].

We​ worked on several problems​‌ pertaining to the design​​ of strict Lyapunov functions​​​‌ i.e. Lyapunov functions having​ a strictly negative-definite derivative​‌ along the trajectories of​​ the studied system.

In​​ 17 and 32,​​​‌ we studied the robust‌ stability for a large‌​‌ class of linear time-varying​​ systems under the assumption​​​‌ that the system possesses‌ some kind of excitation,‌​‌ which is necessary for​​ uniform attractivity of the​​​‌ origin, but not even‌ boundedness of the solutions‌​‌ is assumed a priori.​​ We obtained strict Lyapunov​​​‌ functions, constructed based on‌ an initial candidate whose‌​‌ derivative is sign-undefined. The​​ Lyapunov function that we​​​‌ proposed guarantees input-to-state stability‌ with respect to bounded‌​‌ additive inputs. As a​​ byproduct of our main​​​‌ results, we provided a‌ Lyapunov function for a‌​‌ class of systems reminiscent​​ of model-reference adaptive control​​​‌ with non-differentiable regressors.

In‌ 29, we revisited‌​‌ the problem of designing​​ a strict Lyapunov function​​​‌ for systems appearing in‌ model-reference adaptive control. Most‌​‌ of the results available​​ in the literature are​​​‌ based on the assumption‌ that the regressor is‌​‌ smooth and its derivative​​ admits a known uniform​​​‌ bound. Our main result‌ consisted in relaxing this‌​‌ assumption, which is notably​​ useful in tracking control​​​‌ tasks, where it is‌ required to track arbitrarily‌​‌ rapid reference trajectories. Our​​ main statement was formulated​​​‌ for linear time-varying systems‌ with regressors of dimension‌​‌ one, but we showed​​ how it applies in​​​‌ the analysis of nonlinear‌ systems, notably in adaptive‌​‌ control and adaptive observer​​ design.

In , we​​​‌ studied stability and robustness‌ for a large class‌​‌ of linear time-varying systems​​ under a persistency of​​​‌ excitation assumption, which is‌ necessary for uniform attractivity‌​‌ of the origin. But​​ we did not assume​​​‌ a priori boundedness of‌ the solutions. We obtained‌​‌ strict Lyapunov functions. We​​ performed our construction by​​​‌ using an initial candidate‌ Lyapunov function whose derivative‌​‌ is sign-undefined. The Lyapunov​​ function that we constructed​​​‌ guarantees uniform global asymptotic‌ stability and input-to-state stability‌​‌ with respect to bounded​​ additive inputs. As a​​​‌ byproduct of our main‌ results, we provide a‌​‌ Lyapunov function for a​​ class of systems reminiscent​​​‌ of model-reference adaptive control‌ with non-differentiable regressors.

6.12‌​‌ Local Halanay’s Inequality

Participants:​​ Frédéric Mazenc, Michael​​​‌ Malisoff [LSU].

In‌ the contribition 16,‌​‌ we provided a local​​ version of a stability​​​‌ analysis technique for continuous-time‌ nonlinear systems that is‌​‌ based on the celebrated​​ Halanay’s inequality. The results​​​‌ we obtained ensure local‌ asymptotic stability of an‌​‌ equilibrium point. They are​​ applicable to a family​​​‌ of nonlinear systems that‌ contain state and input‌​‌ delays. We determined input-to-state​​ stability inequalities when the​​​‌ systems contain additive uncertainty.‌ We combined the results‌​‌ with an observer and​​ a Gramian approach, to​​​‌ solve an output feedback‌ stabilization problem. Our numerical‌​‌ examples illustrated how our​​ theorems lead to new​​​‌ basin of attraction estimates.‌

6.13 Event-Triggered Control

Participants:‌​‌ Frédéric Mazenc, Michael​​ Malisoff [LSU].

In​​​‌ the contribution, 33,‌ we proved global exponential‌​‌ stability estimates for a​​ class of nonlinear control​​​‌ systems that contain uncertain‌ time-varying input delays and‌​‌ uncertain state delays. We​​ proved that our new​​​‌ dynamic event-triggered controls ensured‌ that Zeno's phenomenon does‌​‌ not occur. We used​​​‌ new synergies of the​ estimation tools called 'interval​‌ observers' and the celebrated​​ Halanay's inequality to prove​​​‌ the main result. We​ illustrated our approach in​‌ a marine robotic dynamics​​ that contains uncertain nonlinear​​​‌ terms.

6.14 Finite time​ and fixed time stabilization​‌

Participants: Frédéric Mazenc,​​ Michael Malisoff [LSU].​​​‌

In the contribution 34​, we designed sampled​‌ controls to obtained finite-time​​ stabilization properties for continuous-time​​​‌ time-varying linear systems. Our​ main results yielded finite-time​‌ input-to-state stability, where the​​ upper bounding supremum of​​​‌ the uncertainty is over​ a time interval of​‌ constant finite length. Our​​ work included output feedback​​​‌ stabilization and input delays.​ We used our results​‌ to prove novel global​​ exponential input-to-state estimates for​​​‌ nonlinear systems with state​ delays, including systems with​‌ outputs, using a stability​​ analysis technique 'trajectory based​​​‌ approach'. We illustrated our​ work using a pendulum​‌ dynamics with poorly known​​ friction.

In the contribution​​​‌ 35, for discrete-time​ time-varying linear systems, we​‌ constructed sampled-data control laws​​ which result closed-loop systems​​​‌ for which one can​ find fixed-time input-to-state stability​‌ estimates. The upper bounds​​ for the norms of​​​‌ the states in the​ estimates are suprema of​‌ the uncertainties over time​​ intervals of constant finite​​​‌ lengths. We covered output​ feedback stabilization and input​‌ delays. We combined our​​ results with a trajectory​​​‌ based approach, to prove​ novel global exponential input-to-state​‌ stability estimates for nonlinear​​ systems with state delays.​​​‌ We illustrated our work​ using a dynamics containing​‌ an unknown nonlinearity and​​ an unknown state delay.​​​‌

6.15 Lur’e Systems Under​ Incremental-Like Restrictions

Participants: Giorgio​‌ Valmorbida, Gioia Montana​​, Andrea Cristofaro [Università​​​‌ di Roma, La Sapienza]​, Mattia Mattioni [Università​‌ di Roma, La Sapienza]​​.

We look for​​​‌ global asymptotic stability certificates​ for discrete-time systems in​‌  20. These certificates​​ are expressed as Lyapunov​​​‌ functions. The main goal​ is to formulate stability​‌ conditions that are exact,​​ namely do not rely​​​‌ on approximations of terms​ in the stability conditions.​‌ The main target to​​ these stability conditions is​​​‌ their use on the​ analysis iterative algorithms. The​‌ proposed incremental description of​​ the nonlinearities allows us​​​‌ to consider systms with​ continuous but non-differentiable nonlinearities.​‌ The framework is suitable​​ to address multivariate coupled​​​‌ nonlinearities.

6.16 Iterative methods​ to solve Linear Programs​‌

Participants: Giorgio Valmorbida,​​ Morten Hovd [NTNU].​​​‌

The paper 26 presents​ a strategy to solve​‌ an implicit equation obtained​​ from the optimality conditions​​​‌ of linear programs. It​ consists of an iterative​‌ method looking for active​​ sets of constraints where​​​‌ each iteration needs to​ perform the update of​‌ a matrix inversion where​​ a rank two update​​​‌ is introduced. The method​ is applied to Linear​‌ Programs stemming from Model​​ Predictive Control which present​​​‌ a particular structure that​ is exploited in the​‌ algorithm.

6.17 Risk-averse optimization​​

Participants: Laurent Pfeiffer,​​​‌ Riccardo Bonalli [L2S],​ Benoît Bonnet-Weill [L2S].​‌

The preprint 48 establishes​​ a new characterisation of​​​‌ coherent and law-invariant risk​ measures, with the help​‌ of a generalized optimal​​ transport problem. When the​​ associated transport problem is​​​‌ convex, this allows us‌ to demonstrate a novel‌​‌ duality formula for risk​​ measures that is of​​​‌ interest in a numerical‌ perspective.

6.18 Chemical master‌​‌ equation

Participants: Laurent Pfeiffer​​, Guillaume Ballif [Inria​​​‌ Saclay], Jakob Ruess‌ [Inria Saclay].

The‌​‌ preprint 47 is dedicated​​ to continuous-time Markov chains​​​‌ on ${\mathbb{N}}^d$,‌ modelling the evolution of‌​‌ a chemical reaction network​​ (involving $d$ species). We​​​‌ propose a novel bounding‌ technique, involving a Markov‌​‌ chain on $\mathbb{N}$,​​ which is easier to​​​‌ investigate. This allows us‌ to infer in a‌​‌ convenient fashion fundamental properties​​ for the original Markov​​​‌ chain, such as the‌ existence of a stationary‌​‌ distribution.

6.19 Management of​​ fisheries using a mean-field-game​​​‌ approach

Participants: Ziad Kobeissi‌, Idriss Mazari-Fouquer [University‌​‌ Paris Dauphine-PSL], Domenec​​ Ruiz-Balet [University Paris Dauphine-PSL]​​​‌.

We propose a‌ novel model for managing‌​‌ fisheries, described by a​​ system of three coupled​​​‌ partial differential equations. The‌ first is a reaction-diffusion‌​‌ equation representing the dynamics​​ of the fish population,​​​‌ which follows standard approaches‌ in the mathematical literature‌​‌ on spatial ecology. The​​ other two equations are​​​‌ derived using a mean-field-game‌ (MFG) framework to model‌​‌ a large population of​​ fishermen, where the number​​​‌ of fishermen is assumed‌ to be large enough‌​‌ to be treated as​​ infinite. Each fisherman aims​​​‌ to maximize his individual‌ profit, calculated as the‌​‌ revenue from selling fish​​ minus the cost of​​​‌ moving his boat. Under‌ two different structural assumptions‌​‌ about the nonlinearities in​​ the fish dynamics, we​​​‌ prove theoretical results illustrating‌ the tragedy of the‌​‌ commons. Specifically, we show​​ that a lack of​​​‌ coordination among fishermen can‌ significantly harm, or even‌​‌ lead to the extinction​​ of the fish population.​​​‌ Our findings are supported‌ by several numerical simulations.‌​‌

In 65 the main​​ focus is on the​​​‌ derivation of analytical results‌ (e.g existence, uniqueness) and‌​‌ of long time behaviour​​ (here, convergence to the​​​‌ ergodic system) on the‌ mean-field-game system of fishing‌​‌ introduced in 66.​​ We develop novel and​​​‌ versatile tools to address‌ this new class of‌​‌ MFG systems, where agent​​ interactions are indirect and​​​‌ mediated through the solution‌ of a third coupled‌​‌ partial differential equation. The​​ challenging issue of uniqueness,​​​‌ a well-known difficulty in‌ the mean-field game literature,‌​‌ is partially tackled using​​ an innovative approach based​​​‌ on spectral methods inspired‌ by shape optimization theory.‌​‌ We establish the convergence​​ of the time-dependent system​​​‌ to its ergodic counterpart‌ as the time horizon‌​‌ tends to infinity, both​​ in the first- and​​​‌ second-order cases.

6.20 Prescribed‌ stabilization and autoregressive control‌​‌ in vibration damping

Participants:​​ Islam Boussaada, Silviu​​​‌ Niculescu, Sami Tliba‌ [L2S].

As an‌​‌ application of the partial​​ poles placement in vibration​​​‌ damping, we used in‌ 46 the autoregressive control‌​‌ which has sufficient parametric​​ degrees of freedom available​​​‌ for the controller's design.‌ It turns out that‌​‌ the use of dynamical​​ parameters corresponds to linear​​​‌ filtered terms in the‌ control law of the‌​‌ original one. To emphasize​​​‌ the benefits brought by​ such a controller, the​‌ active vibration damping problem​​ is addressed for a​​​‌ flexible mechanical structure equipped​ with a collocated pair​‌ of piezoelectric sensor and​​ actuator.

6.21 Control of​​​‌ DC/DC converters

Participants: Frédéric​ Mazenc, Alessio Iovine​‌ [L2S].

The paper​​ 31 falls within the​​​‌ research area of advanced​ control of power converters.​‌ In it, we proposed​​ a bounded control strategy​​​‌ for a nonlinear model​ of a DC/DC boost​‌ converter that guarantees both​​ boundedness of all closed-loop​​​‌ signals and stability of​ the overall system. We​‌ provided explicit guidelines for​​ selecting the controller parameters.​​​‌ We performed simulations to​ validate the effectiveness of​‌ the proposed control approach​​ in achieving reliable voltage​​​‌ regulation of the converter.​

6.22 A new tool​‌ for the simulation of​​ controllers for retarded delay​​​‌ systems

Participants: Catherine Bonnet​, Duc Duy Do​‌, Mustafa Oguz Yegin​​ [Bilkent University], Hitay​​​‌ Ozbay [Bilkent University].​

We developed a new​‌ software tool for reliable​​ implementation of all stabilizing​​​‌ controllers for general retarded​ time delay systems, and​‌ the case of systems​​ with one delay in​​​‌ the numerator and one​ delay in the denominator​‌ was presented in 25​​. This work was​​​‌ the continuation of 56​; it demonstrated a​‌ Matlab/Simulink realization and how​​ users interface with it.​​​‌ We also discussed some​ technical issues in the​‌ implementation of the infinite​​ dimensional parts of the​​​‌ controller blocks.

7.1 Bilateral contracts​​ with industry

Participants: Laurent​​​‌ Pfeiffer, François Caffier​.

The team signed​‌ in 2024 a contract​​ with IFPEN in the​​​‌ framework of the Inria-IFPEN​ contract 20 June 2020.​‌ A PhD is funded​​ for 3 years.

8.1​ International initiatives

8.2 International​​​‌ research visitors

8.3 National initiatives‌

• L. Pfeiffer is involved‌​‌ in the EDF-Inria DEFI​​ on the "management of​​​‌ tomorrow's electrical systems".

9.1 Promoting scientific‌​‌ activities

9.2​‌ Teaching - Supervision -​​ Juries - Educational and​​​‌ pedagogical outreach

9.3‌ Popularization

10.1 Publications of‌ the year

10.2 Cited​​​‌ publications

• 53 articleJ.​Jean Auriol and F.​‌Florent Di Meglio.​​ An explicit mapping from​​​‌ linear first order hyperbolic​ PDEs to difference systems​‌.Systems Control Lett.​​1232019, 144--150​​​‌URL: https://doi.org/10.1016/j.sysconle.2018.11.012DOIback​ to text
• 54 article​‌C. E.Cerino E.​​ Avellar and J. K.​​​‌Jack K. Hale.​ On the zeros of​‌ exponential polynomials.J.​​ Math. Anal. Appl.73​​​‌21980, 434--452​URL: https://doi.org/10.1016/0022-247X(80)90289-9DOIback​‌ to textback to​​ text
• 55 articleC.​​​‌ E.C. E. de​ Avellar and S. A.​‌S. A. S. Marconato​​. Difference equations with​​​‌ delays depending on time​.Bol. Soc. Brasil.​‌ Mat. (N. S.)21​​11990, 51--58​​​‌URL: https://doi.org/10.1007/BF01236279DOIback​ to text
• 56 inproceedings​‌C.Catherine Bonnet,​​ H.Hitay Özbay,​​​‌ M. O.Mustafa Oguz​ Yegin and S.Suat​‌ Gumussoy. Implementation of​​ Stabilizing Controllers for Retarded​​​‌ Delay Systems.IFAC​ papers Online5827​‌Dimitri BredaUdine, Italy​​September 2024, 84-89​​​‌HALDOIback to​ text
• 57 inproceedingsI.​‌Islam Boussaada, G.​​Guilherme Mazanti, S.-I.​​​‌Silviu-Iulian Niculescu, J.​Julien Huynh, F.​‌Franck Sim and M.​​Matthieu Thomas. Partial​​​‌ Pole Placement via Delay​ Action: A Python Software​‌ for Delayed Feedback Stabilizing​​ Design.2020 24th​​​‌ International Conference on System​ Theory, Control and Computing​‌ (ICSTCC)2020 24th International​​ Conference on System Theory,​​​‌ Control and Computing (ICSTCC)​Sinaia, RomaniaOctober 2020​‌, 196--201HALDOI​​back to text
• 58​​​‌ articleI.Islam Boussaada​, G.Guilherme Mazanti​‌ and S.-I.Silviu-Iulian Niculescu​​. The generic multiplicity-induced-dominancy​​ property from retarded to​​​‌ neutral delay-differential equations: When‌ delay-systems characteristics meet the‌​‌ zeros of Kummer functions​​.Comptes Rendus. Mathématique​​​‌3602022, 349--369‌HALDOIback to‌​‌ text
• 59 articleR.​​ K.Robert K. Brayton​​​‌. Bifurcation of periodic‌ solutions in a nonlinear‌​‌ difference-differential equations of neutral​​ type.Quart. Appl.​​​‌ Math.241966,‌ 215--224URL: https://doi.org/10.1090/qam/204800DOI‌​‌back to text
• 60​​ articleY.Yacine Chitour​​​‌, G.Guilherme Mazanti‌ and M.Mario Sigalotti‌​‌. Stability of non-autonomous​​ difference equations with applications​​​‌ to transport and wave‌ propagation on networks.‌​‌Netw. Heterog. Media11​​42016, 563--601​​​‌URL: https://doi.org/10.3934/nhm.2016010DOIback‌ to textback to‌​‌ text
• 61 articleK.​​ L.Kenneth L. Cooke​​​‌ and D. W.David‌ W. Krumme. Differential-difference‌​‌ equations and nonlinear initial-boundary​​ value problems for linear​​​‌ hyperbolic partial differential equations‌.J. Math. Anal.‌​‌ Appl.241968,​​ 372--387URL: https://doi.org/10.1016/0022-247X(68)90038-3DOI​​​‌back to text
• 62‌ bookJ. K.Jack‌​‌ K. Hale and S.​​ M.Sjoerd M. Verduyn​​​‌ Lunel. Introduction to‌ functional-differential equations.99‌​‌Applied Mathematical SciencesSpringer-Verlag,​​ New York1993,​​​‌ x+447URL: https://doi.org/10.1007/978-1-4612-4342-7DOI‌back to textback‌​‌ to textback to​​ textback to text​​​‌back to text
• 63‌ articleJ. K.Jack‌​‌ K. Hale and S.​​ M.Sjoerd M. Verduyn​​​‌ Lunel. Strong stabilization‌ of neutral functional differential‌​‌ equations.IMA J.​​ Math. Control Inform.19​​​‌1-2Special issue on‌ analysis and design of‌​‌ delay and propagation systems​​2002, 5--23URL:​​​‌ https://doi.org/10.1093/imamci/19.1_and_2.5DOIback to‌ text
• 64 articleD.‌​‌Daniel Henry. Linear​​ autonomous neutral functional differential​​​‌ equations.J. Differential‌ Equations151974,‌​‌ 106--128URL: https://doi.org/10.1016/0022-0396(74)90089-8DOI​​back to textback​​​‌ to text
• 65 article‌Z.Ziad Kobeissi,‌​‌ I.Idriss Mazari-Fouquer and​​ D.Domènec Ruiz-Balet.​​​‌ Mean-field games for harvesting‌ problems: Uniqueness, long-time behaviour‌​‌ and weak KAM theory​​.Journal of Differential​​​‌ Equations4482025,‌ 113667back to text‌​‌
• 66 articleZ.Ziad​​ Kobeissi, I.Idriss​​​‌ Mazari-Fouquer and D.Domènec‌ Ruiz-Balet. The tragedy‌​‌ of the commons: A​​ Mean-Field Game approach to​​​‌ the reversal of travelling‌ waves.Nonlinearity37‌​‌112024, 115010​​back to text
• 67​​​‌ bookW.Wim Michiels‌ and S.-I.Silviu-Iulian Niculescu‌​‌. Stability, control, and​​ computation for time-delay systems​​​‌.27Advances in‌ Design and ControlAn‌​‌ eigenvalue-based approachSociety for​​ Industrial and Applied Mathematics​​​‌ (SIAM), Philadelphia, PA2014‌, xxiv+435URL: https://doi.org/10.1137/1.9781611973631‌​‌DOIback to text​​
• 68 phdthesisR. A.​​​‌Richard Andrew Silkowski.‌ Star-shaped regions of stability‌​‌ in hereditary systems.​​Division of Applied Mathematics,​​​‌ Brown University1976back‌ to textback to‌​‌ text