The goal of the project is to better understand and well formalize the effects of complex environments on the dynamics of the interconnections, as well as to develop new methods and techniques for the analysis and control of such systems.

It is well-known that the interconnection of dynamic systems has as consequence an increased complexity of the behavior of the total system.

In a simplified way, as the concept of dynamics is well-understood, the interconnections can be seen as associations (by connections of materials or information flows) of distinct systems to ensure a pooling of the resources with the aim of obtaining a better operation with the constraint of continuity of the service in the event of a fault. In this context, the environment can be seen as a collection of elements, structures or systems, natural or artificial constituting the neighborhood of a given system. The development of interactive games through communication networks, control from distance (e.g. remote surgical operations) or in hostile environment (e.g. robots, drones), as well as the current trend of large scale integration of distribution (and/or transport and/or decision) and open information systems with systems of production, lead to new modeling schemes in problems where the dynamics of the environment have to be taken into account.

In order to tackle the control problems arising in the above examples, the team investigates new theoretical methods, develops new algorithms and implementations dedicated to these techniques.

The major questions considered are those of the characterization of the stability (also including the problems of sensitivity compared to the variations of the parameters) and the determination of stabilizing controllers of interconnected dynamic systems. In many situations, the dynamics of the interconnections can be naturally modelled by systems with delays (constant, distributed or time-varying delays) possibly of fractional order. In other cases, partial differential equations (PDE) models can be better represented or approximated by using systems with delays. Our expertise on this subject, on both time and frequency domain methods, allows us to challenge difficult problems (e.g. systems with an infinite number of unstable poles).

Robust stability of linear systems

Within an interconnection context, several phenomena are modelled directly or after an approximation by delay systems. These systems may have constant delays, time-varying delays, distributed delays.

For various infinite-dimensional systems, particularly delay and fractional systems, input-output and time-domain methods are jointly developed in the team to characterize stability. This research is developed at four levels: analytic approaches (

Robustness/fragility of biological systems

Deterministic biological models describing, for instance, species interactions, are frequently composed of equations with important disturbances and poorly known parameters. To evaluate the impact of the uncertainties, we use the techniques of design of global strict Lyapunov functions or functional developed in the team.

However, for other biological systems, the notion of robustness may be different and this question is still in its infancy (see, e.g. 89). Unlike engineering problems where a major issue is to maintain stability in the presence of disturbances, a main issue here is to maintain the system response in the presence of disturbances. For instance, a biological network is required to keep its functioning in case of a failure of one of the nodes in the network. The team, which has a strong expertise in robustness for engineering problems, aims at contributing at the development of new robustness metrics in this biological context.

Linear systems: Analytic and algebraic approaches are considered for infinite-dimensional linear systems studied within the input-output framework.

In the recent years, the Youla-Ku

A central issue studied in the team is the computation of such factorizations for a given infinite-dimensional linear system as well as establishing the links between stabilizability of a system for a certain norm and the existence of coprime factorizations for this system. These questions are fundamental for robust stabilization problems 2, 3.

We also consider simultaneous stabilization since it plays an important role in the study of reliable stabilization, i.e. in the design of controllers which stabilize a finite family of plants describing a system during normal operating conditions and various failed modes (e.g. loss of sensors or actuators, changes in operating points). Moreover, we investigate strongly stabilizable systems, namely systems which can be stabilized by stable controllers, since they have a good ability to track reference inputs and, in practice, engineers are reluctant to use unstable controllers especially when the system is stable.

Nonlinear systems

In any physical systems a feedback control law has to account for limitation stemming from safety, physical or technological constraints. Therefore, any realistic control system analysis and design have to account for constraints appearing mainly from sensors and actuators nonlinearities and from the regions of safe operation in the state space. This motivates the study of linear systems with more realistic, thus complex, models of actuators accounting for saturation and quantization at the inputs of the system 17, 12, 16.

The project aims at developing robust stabilization theory and methods for important classes of nonlinear systems that ensure good controller performance under uncertainty and time delays. The main techniques include techniques called backstepping and forwarding, contructions of strict Lyapunov functions through so-called "strictification" approaches 8 and construction of Lyapunov-Krasovskii functionals 9, 10, 11 or or Lyapunov functionals for PDE systems 15.

PID controllers

Even though the synthesis of control laws of a given complexity is not a new problem, it is still open, even for finite-dimensional linear systems. Our purpose is to search for good families of “simple” (e.g. low order) controllers for infinite-dimensional dynamical systems. Within our approach, PID candidates are first considered in the team 3, 91.

For interconnected systems appearing in teleoperation applications, such as the steer-by-wire, Proportional-Derivative laws are simple control strategies allowing to reproduce the efforts in both ends of the teleoperation system. However, due to delays introduced in the communication channels these strategies may result in loss of closed loop stability or in performance degradation when compared to the system with a mechanical link (no communication channel). In this context we search for non-linear proportional and derivative gains to improve performance 6, 5. This is assessed in terms of reduction of overshoot and guaranteed convergence rates.

Delayed feedback

Control systems often operate in the presence of delays, primarily due to the time it takes to acquire the information needed for decision-making, to create control decisions and to execute these decisions. Commonly, such a time delay induces desynchronizing and/or destabilizing effects on the dynamics. However, some recent studies have emphasized that the delay may have a stabilizing effect in the control design. In particular, the closed-loop stability may be guaranteed precisely by the existence of the delay. The interest of considering such control laws lies in the simplicity of the controller as well as in its easy practical implementation. It is intended by the team members to provide a unified approach for the design of such stabilizing control laws for finite and infinite dimensional plants 4, 13.

Finite Time and Interval Observers for nonlinear systems

We aim to develop techniques of construction of output feedbacks relying on the design of observers. The objectives pertain to the design of robust control laws which converge in finite time, the construction of intervals observers which ensure that the solutions belong to guaranteed intervals, continuous/discrete observers for systems with discrete measurements and observers for systems with switches.

Finally, the development of algorithms based on both symbolic computation and numerical methods, and their implementations in dedicated Scilab/Matlab/Maple toolboxes are important issues in the project.

The team is involved in life sciences applications. The two main lines are the analysis of bioreactors models (microorganisms; bacteria, microalgae, yeast, etc..) and the modeling of cell dynamics in Acute Myeloblastic Leukemias (AML) in collaboration with St Antoine Hospital in Paris.

The team is interested in Energy management and considers control problems in energy networks.

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.

The team is interested in vibration control (in link with the so-called multiplicity-induced-dominancy, MID and partial pole placement) and in developing advanced delay algorithms for compensating and tracking periodic signals (related to the repetitive control).

YALTAPy is a Python Toolbox dedicated to the stability analysis of (fractional) delay systems given by their transfer function.

The delays are supposed to be commensurate.

In the case of retarded systems or of neutral systems with asymptotic axes in the left half-plane, YALTAPy gives: – For a given delay , the number and the position of unstable poles. – For which values of the delay the system is stable, – For a set of values of the delay, the position of unstable poles (root locus).

The YALTAPy toolbox is a Python toolbox dedicated to the study of classical and fractional systems with delay in the frequency-domain. Its objective is to provide basic but important information such as, for instance, the position of the neutral chains of poles and unstable poles, as well as the root locus with respect to the delay of the system.

YALTAPy_Online is an online version of YALTAPy

In two works, we have have developed a new avaraging technique for fast-varying continuous-time systems.

In 54, we studied linear systems with fast-varying almost periodic coefficients. We presented a novel transformation of the fast varying coefficients. This transformation enabled us to perform averaging over multiple time-scales for systems with constant delays, where the value of delay is not small (i.e. arbitrarily large with respect to the small parameter). We carried out stability analysis by employing time-varying Lyapunov functions (or functionals for the delayed case). The analysis leads to LMI conditions that are always feasible for small enough parameters.

The paper 53 is devoted to the problem of establishing input-to-state stability (ISS) for linear systems with multiple time-scales. The considered systems contain rapidly-varying, piecewise continuous and almost periodic coefficients with small parameters. The stability analysis relies on a novel system transformation, leading to a new system whose ISS guarantees the ISS of the original one. In this work, we unified this transformation with a new superposition-based system presentation. We employed time-varying Lyapunov functions for ISS analysis, where the novel system presentation plays a crucial role in deriving essentially less conservative compensating upper bounds. The analysis yields conditions of LMI type for ISS, leading to explicit bounds on the small parameters, decay rate and ISS gains. The LMIs are accompanied by suitable feasibility guarantees.

Event-triggered control has the advantage that it can reduce computational burdens of implementing feedback controls, by only changing control values when a significant enough event occurs. In order to decrease the number of needed switches of the control laws, we developed results relying on the theory of the positive systems and comparison systems called interval observers. In the papers 37 and 56, we provided a new input-to-state stabilizing event-triggered feedback design for linear systems with unknown input delays, unknown measurement delays, and unknown additive disturbances. We used the theory of positive systems, interval observers, and a vector version of Halanay’s inequality. We illustrated our method using a marine robotic model.

In contribution 39, to help the analysis of the stability properties of discrete-time systems, we provided a discrete-time vector valued analog of recently developed continuous-time trajectory-based estimates. We used it to provide a discrete-time version of the celebrate Halanay’s inequality. We combined the results with interval observers to prove exponential stability properties for discrete-time linear systems with uncertainties whose arbitrarily long input delays are compensated for by reduction model controls, and a robust global exponential stability result for observers for discrete-time linear systems.

In the papers 38 and 57, we provided new observer designs to simultaneously identify parameters and states of systems whose non-linearities have order two near the origin, which include cubic terms arising in the study of jump phenomena, process control, and bistable models of aerospace systems. This yields local exponential convergence of the state estimation error to zero, basin of attraction estimates, and fixed time parameter identification. We illustrated our result using Duffing’s equation, whose cubic term puts it outside the scope of prior methods

Piece-wise affine systems appear when linear dynamics are defined in different partitions of the state space. This type of systems naturally appears whenever actuators have different stages or saturate or whenever non-linear control laws are obtained as the solution to a parameterised optimization problem as, for instance for systems with feedback laws based on the so-called explicit Model Predictive Control. Even though the dynamics is simple to describe, the stability analysis, performance assessment and robustness analysis are difficult to perform since, due to the often used explicit representation, the Lyapunov stability and dissipation tests are often described in terms of a number of inequalities that increases exponentially on the number of sets in the partition since they are based on the enumeration of the partition transitions. Moreover regional stability and uncertainties corresponding to modification on the partition are difficult to study in this scenario.

To overcome these difficulties we have proposed an implicit representation for this class of systems in terms of ramp functions. The main advantage of such a representation lies on the fact that the ramp function can be exactly characterized in terms of linear inequalities and a quadratic equation, namely a linear complementarity condition. Thanks to the characterization of the ramp function and the implicit description of the PWA system, the verification of Lyapunov inequalities related to piecewise quadratic functions can be formulated as a semidefine programming whenever some co-positivity constraints are relaxed.

We have applied the results to the local analysis and synthesis of PWA control laws. Such a local formulation is based on local conditions for co-positivity of matrices. The proposed results encompass regional stability analysis formulations in the literature.

The stability conditions rely on the solution via convex optimization of piecewise quadratic inequalities in an implicit form, which can also be used to compute lower bounds to the minimum of non-convex and discontinuous functions.

A novel method is proposed for solving quadratic programming problems arising in model predictive control. The method is based on an implicit representation of the Karush–Kuhn–Tucker conditions using ramp functions. The method is shown to be highly efficient on both small and fairly large Quadratic Program problems, can be implemented using simple computer code, and has modest memory requirements. The proposed algorithm shows a performance improvement mainly for large dimension optimization problems whenever few constraints are active in the optimal solution.

One-dimensional hyperbolic systems are useful models for a wide range of natural phenomena, in particular those involving the propagation of some physical quantities, such as the propagation of electricity along a transmission line, of laser signals in optical fibers, of water in open channels, of water or gaz in ridig pipes, of vehicles in a road system, of vibrations on mechanical structures, of living organisms in the presence of certain chemicals, among many others 84. These numerous applications have motivated many recent works on the analysis of these kinds of systems, such as 83, 84, which address questions such as their stability, stabilizability, and controllability.

In the recent work 74, we have obtained criteria for exact and approximate controllability of linear one-dimensional hyperbolic systems in

Finally, 74 considers the special case of flows in networks, identifying a topological obstruction for controllability and deducing Hautus-type controllability criteria for these systems, which reduce to Kalman criteria in the case where the delays of the corresponding difference equation are commensurate.

The dynamics of systems in which phenomena of different nature are present may contain different time scales. This is the case, for instance, of electric motors, in which the electrical time scale is typically much faster than the mechanical one. A natural question in these situations, which is the basis of the singular perturbation theory, is whether one can approximate the fastest time scale by an instantaneous process. For finite-dimensional systems, its answer is given by Tikhonov's theorem, which states roughly that such an approximation is valid as soon as the dynamics of the fastest time scale is stable 90.

However, many systems in practice involving propagative, diffusive, or reactive phenomena are naturally modeled as infinite-dimensional systems. Singular perturbation in infinite dimension has attracted much interest from researchers in recent years, and it was highlighted in particular that classical results for finite-dimensional systems may fail to hold true in general in such a context, according to which system evolves in a fast time scale (see, e.g., 92). Most works in the literature on singular perturbation for infinite-dimensional systms deal only with specific systems, showing approximation properties, when they hold, through the use of Lyapunov functions, as done in 92.

In order to get a better understanding of when singular perturbation methods work well for infinite-dimensional systems, we have revisited, in the recent work 50, the system considered in 92 made of a slow ODE coupled with a fast one-dimensional transport PDE. By adopting a spectral point of view of the problem, we have obtained sharper conditions under which approximation properties hold true, improving the results of 92. In addition, the spectral methods used in 50 have provided valuable insight on the behavior of singularly perturbed systems in infinite dimension, which might be used to address more general systems than the one from that reference.

Since the seminal works 86, 87, it has been known that spectral values of some families of time-delay systems of large multiplicity are often dominant (i.e., they attain the spectral abcissa of the system, and determine thus its asymptotic behavior), a property which came to be known as the multiplicity-induced-dominancy (MID) property. The validity of this property has an important impact in the stabilization of time-delay systems, since, when it holds, one may stabilize a system by selecting its free parameters in order to ensure that it admits a spectral value of large multiplicity which is dominant and has negative real part, ensuring stability.

In 72, we have studied the MID property for a family of delay-differential equations with a single delay when considering spectral values of multiplicity equal to

The work 73 extends the results of 72 to roots of multiplicity larger than or equal to control-oriented configuration, in which the coefficients of highest order of the non-delayed part of the system are fixed, and all others are free. Sufficient conditions for dominance, similar to those of 72, are also provided in 73, together with a detailed analysis of two examples.

For instance, in 19, we characterize the MID property in the scalar neutral case with respect to the system parameters. Particular attention is paid to the so-called over-order multiplicities corresponding to real double and triple characteristic roots.

While most results on the MID property consider only systems with a single delay, the work 31 has addressed a first-order system with two delays. By considering the ratio between the smallest and the largest delay as a parameter, 31 provides a careful analysis of the behavior of the spectrum of the system with respect to this parameter, which is used to establish the MID property for roots of maximal multiplicity of such a class of systems. As a consequence, 31 also establishes that the inclusion of a second delay may help in stabilizing time-delay systems with constraints on their coefficients, with respect to a classical proportional-delayed controller. As an extension of the methodology of the latter, 59 characterizes the MID property for second-order systems controlled by a two-delay "block". As an application of which, the problem of stabilization of the classical pendulum with exclusive access to the delayed position is treated.

In recent years, the Team highligted an extension of the MID property called coexistent-real-roots-induced-dominancy (CRRID), see for instance 85, 82 applying for LTI functional differential equations of retarded type. The CRRID property consists in conditions on an LTI functional dynamical system's parameters guaranteeing the dominancy of coexistence of real spectral values.
In 45, we extend such a property to a class of neutral systems, and exploit it in the boundary control of the standard transport equation. Namely, by using the CRRID property, we show that one can arbitrarily and robustly prescribe the exponential decay of the closed-loop transport solution, yielding the prospect of applying the CRRID partial poles placement methodology to hyperbolic PDE's.

We deal in 18 with a control-affine problem with scalar control subject to bounds, a scalar state con- straint and endpoint constraints of equality type. For the numerical solution of this problem, we propose a shooting algorithm and provide a sufficient condition for its local convergence. We exhibit an example that illustrates the theory.

We investigate in 75 the convergence of the Generalized Frank-Wolfe (GFW) algorithm for the resolution of potential and convex second-order mean field games. In a previous work 34, we had established some rates of convergence for this method, at the continuous level. We analyze here the impact of the discretization of the mean-field-game system on the effectiveness of the GFW algorithm. The article focuses on the theta-scheme, which we introduced in 25. A sublinear and a linear rate of convergence are obtained, for two different choices of stepsizes. These rates have the mesh-independence property: the underlying convergence constants are independent of the discretization parameters.

In the preprint 76, we formulate and investigate a mean field optimization (MFO) problem over a set of probability distributions

In the paper 32, we derived feedback control laws for isolation, contact regulation, and vaccination for infectious diseases, using a strict Lyapunov function. We use an SIQR (Susceptible, Infected, Quarantined, Recovered individuals) epidemic model describing transmission, isolation via quarantine, and vaccination for diseases to which immunity is long-lasting. Assuming that mass vaccination is not available to completely eliminate the disease in a time horizon of interest, we provided feedback control laws that drive the disease to a small endemic equilibrium. We prove the ISS robustness property on the entire state space, when the immigration perturbation is viewed as the uncertainty. We use an ISS Lyapunov function to construct the feedback control laws. A key ingredient in our analysis is that all compartment variables are present not only in the Lyapunov function, but also in a negative definite upper bound on its time derivative. We illustrate the efficacy of our method through simulations. Since the control laws are feedback, their values are updated based on data acquired in real time. We also discussed the degradation caused by the delayed data acquisition occurring in practical implementations, and we derive bounds on the delays under which the ISS property is maintained when delays are present.

An internal model control scheme is proposed in 49 to compensate both a long dead-time of a system and a harmonic disturbance. The controller is based on an inversion of the first-order model used to approximate the system dynamics together with an input delay. Two other components of the controller consist of a filter and an additional delay by which the harmonic modes are targeted via adjusting the control loop gain and phase shift. The design of the filter-delay pair is fully analytical and the implementation of the scheme is straightforward. The main attention is paid to the complete compensation of a single harmonic disturbance. Besides, an extension of the scheme is proposed to target a double harmonic disturbance. Increased attention is paid to the robustness aspects of the schemes. Outstanding performance in terms of harmonic disturbance compensation of the proposed schemes is demonstrated on a series of laboratory experiments.

The paper 48 presents a controller design for systems suffering from multi-harmonic periodic disturbance and substantial input time-delay. It forms an alternative approach to Repetitive Control where the goal is to stabilize a closed-loop that encapsulates an explicit time-delay model of the periodic signal. The proposed controller design is based on the Internal Model Control (IMC) framework, and it consists of the inverse system model and an appropriate distributed delay with an overall length related to the period of the disturbance. The properness of the controller can be ensured by utilizing a low-pass filter, however, such a component is shown to be unnecessary when the relative order of the system model is one. This fact makes the alternative approach especially suitable for systems approximated by a first-order model with input time-delay, leading to a straightforward controller design thanks to its simple structure and attainable conditions. Stability of the configuration is guaranteed by an ideal IMC framework. For further performance and robustness requirements for the non-ideal case the tuning of the controller is posed as a weighted-

Following our project on the modeling and analysis of healthy and unhealthy cell population dynamics in leukemia, we have considered a nonlinear system with distributed delays where the parameters depend on growth-factor concentrations. Here, a change in one of the growth factor concentrations may lead to a switch in the corresponding model parameter. We have achieved a network representation of the switching system involving nodes and edges. Each node stands for a full-fledged nonlinear system with distributed delays where the parameters are constant. For each node, a stable positive steady state may exist. In this network framework, a change in the growth-factor concentration is interpreted as a transition from one node to another. We have proposed a method which provides a (sub)optimal therapeutic strategy, guiding the density of cells from an abnormal state towards a healthy one, through multiple drug infusions.

C. Bonnet is member of the ANR Dreamy

From September 2021 - September 2025

A key advantage of biological computing devices is their ability to sense, compute, and especially to respond to their biological environment, e.g., bacteria can be programmed to act as autonomous robots within the human body. Local presence of certain molecules in the environment allows sensing of neighboring cell types and acting accordingly, e.g., by activating an immune response. Current designs of synthetic circuits in bacteria, however, face severe resource limitations: each genetic part added to the cell imposes an additional burden, becoming progressively toxic for the cell. The most common design techniques for biological logic gates rely on gene regulation via DNA-binding proteins, nucleic acid (DNA/RNA) interactions, or more recently the CRISPR machinery. Each comes with its own constraints: like limited availability of orthogonal signals for use within the cell (DNA-binding), small dynamic range (RNA-based), or reduced growth rates (the CRISPR machinery). This has led to recent efforts to distribute circuits among several cells to reduce the resource load per cell, taking the formative steps towards distributed bacterial circuits. The DREAMY research project seeks to develop innovative solutions to the problem of building distributed circuits in bacteria from an algorithmic, theoretical perspective that contributes to real-world implementable solutions.

Involved groups: LMF (FR), LISN (FR), DISCO (FR), ALGO group (University of Geneva, CH), Micalis (INRAE, FR), L2S (FR)

S-I Niculescu is the Founding Editor and the Editor-in-Chief of the Springer-Nature book series « Advances in delays and dynamics » since its creation in 2012.

S-I Niculescu was the General Co-Chair of the 21st European Control Conference, Bucharest, Romania, June 13-16th, 2023 (ECC 2023: ).

Members of the team have reviewed papers for several journals covering the topics of the team, including the European Control Conference and the IFAC World Congress.

Members of the team have reviewed papers for several journals covering the topics of the team, including Acta Applicandae Mathematicae, Applied Mathematics and Optimization, Automatica, Communications in Contemporary Mathematics, IEEE Transactions on Automatic Control, IMA Journal of Mathematical Control and Information, Journal of Optimization Theory and Applications, and Systems & Control Letters

S-I Niculescu was the Chair of the IFAC Technical Committee Linear Control Systems (TC 2.2; 2017-2023; 171 members).

C. Bonnet was a member of the PhD committees of:

- Z. Wang, 18 Januray 2023, INSA Val de Loire. Title of the thesis: Modulating function-based non-asymptotic and robust estimations for fractional-order systems.

- A. Diab 18 April 2023, Centralesupelec. Title of the thesis: Stability analysis and control design for time-delay systems with applications to automotive steering systems.

- H. Li, 28 June 2023, Université de Lille. Title of the thesis: Cosserat-Based Modeling and Control of Slender Soft Robots.

- K. Saidi, 29 September 2023, Université de Lorraine. Title of the thesis: Stabilisation d’une classe d’EDPs non linéaires. Application à l’équation de Vlasov-Poisson.

- E. Paiva, 20 December 2023, Mines Paris PSL. Title of the thesis: Wind Velocity Estimation for Wind Farms.

- S. Mohite, 20 December 2023, Université de Lorraine. Title of the thesis: Observer Design for Nonlinear Systems using LMI Relaxation Techniques.

She also was a member of the Habilitation committee of M. Di Loreto, 5 October 2023, INSA de Lyon. Title of the habilitation thesis: Propriétés structurelles des systèmes dynamiques pour le contrôle.

S-I Niculescu participated in the launch of the « Cahiers de l’Institut Pascal »(published by EDP Sciences Paris) and he is one of the Associate Editors. This is an initiative launched at the University of Paris-Saclay - a collection of white papers (free access) from thematic programs (2 per year) organized at the institute. The first issue (Physics) was published in 2023, and the evaluation of second one (devoted to « urban mobility ») was finalized November 2023.