2024Activity 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 conntrol 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 $L^p$ (in particular $L^{\infty }$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 ${\displaystyle p_0\left(s\right)+\sum _{i=1}^n{p}_i\left(s\right)e^{-h_{i}s}}$ where $p_{i}i=0,\cdots ,n$ are real polynomials in the complex variable $s$ satisfying $deg{p}_0\ge deg{p}_i$ and $h_i>0$.
• The stability properties of linear delay systems with time-varying delays and/or time varyning 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

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 Control 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

  $$\begin{array}{ccc}\hfill x[k+1]& =& F_{1}x\left[k\right]+F_2\varphi \left(y\left(x\left[k\right]\right)\right)\hfill \\ \hfill y\left(x\right[k\left]\right)& =& F_{3}x\left[k\right]+F_4\varphi \left(y\left(x\left[k\right]\right)\right)+f_5\hfill \end{array}$$

  where the vector-valued function $\varphi :{ℝ}^m\to {ℝ}^m$ 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:

• QSome 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)

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

S.I Niculescu received the IFAC French NMO Award (annual prize which recognizes a personality who has made fundamental contributions to IFAC and its functioning).

6.1 Stability analysis for linear systems with a switched rapidly varying delay

Participants: Frédéric Mazenc, Silviu Niculescu, Diego Torres-García .

In 38, we have studied linear continuous-time systems with switched pointwise delays. We have developed a technique enabling to establish the stability of these systems with a rapidly varying periodic delay. It relies on two ingredients: a representation of the systems as time-varying systems with constant delays and an averaging approach. Illustrative examples showed the effectiveness of the proposed methodology.

6.2 Event-triggered control

Participants: Frédéric Mazenc, Michael Malisoff [LSU], Corina Barbalata [LSU], Safeyya Alyahia [LSU].

In the contribution 23, we provided new dynamic event-triggered controls for continuous-time linear systems that contain additive uncertainties. We established input-to-state stability properties that imply uniform global exponential stability when the additive uncertainties are zero. Significant novel features include (a) new dynamic extensions and new trigger rules that provide a new positive systems analog of significant prior dynamic event-triggered work of A. Girard and (b) the application to a BlueROV2 underwater vehicle model, where we provided significantly larger lower bounds on the inter-execution times, and usefully fewer trigger times, compared with standard dynamic event-triggered approaches that used the usual Euclidean norm, and as compared with static event-triggered controls that instead used positive systems approaches.

6.3 New developments on the multiplicity-induced-dominancy property

Participants: Islam Boussaada [DISCO], Guilherme Mazanti [DISCO], Wim Michiels [K.U. Leuven], Silviu-Iulian Niculescu [DISCO].

The multiplicity-induced-dominancy (MID) property consists on the fact that, if a given time-delay system admits a spectral value of “large enough” multiplicity, then this spectral value is dominant (i.e., the rightmost one in the spectrum) and thus determines the asymptotic behavior of the system. 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 exponential stability. Since the seminal works 92, 93, the MID property has been established for some classes of time-delay systems and has been applied to many contexts 111, 91, and is an active research topic of the DISCO team. During the past year, several new developments on the MID property were established in the team.

The focus of 26 is the MID property for delay-differential equations (DDEs) with a single delay and spectral values with overorder multiplicity, i.e., a multiplicity larger than the order of the DDE. More precisely, that article considered the particular case of retarded DDEs in which the non-delayed part is of order $n$, the delayed part is of order $n-1$, and one is interested in roots of multiplicity at least $n+1$. We have proved that a root of overorder multiplicity is necessarily a root of a particular polynomial, called the elimination-produced polynomial, and addressed the MID property using a suitable factorization of the corresponding characteristic function involving special functions of Kummer type. Additional results and discussion were provided in the case of the $n$th order integrator, in particular on the local optimality of a multiple root. The derived results showed how the delay can be further exploited as a control parameter and were applied to some problems of stabilization of standard benchmarks with prescribed exponential decay.

A generalization of the results of 26 to the case of single-delay DDEs was done in 27. For a single-delay DDE with non-delayed part of order $n$ and delayed part of order $m\le n$, that reference characterized the existence of spectral values of multiplicities between $n+1$ and $n+m+1$ in terms of a suitable integral factorization, which can be rewritten as a linear combination of Kummer functions, the number of Kummer functions appearing in the combination being the difference between the degree of the corresponding quasipolynomial and the multiplicity of the spectral value. We have also considered stabilizability issues with an arbitrary configuration of known and unknown parameters of the system, proving that a root of overorder multiplicity is necessarily a root of a particular function, called the elimination-produced function, which extends the previous notion of elimination-produced polynomial. Theoretical results of the paper were concluded by a sufficient condition for validity of the MID property, which relies on the Green–Hille transformation from 107. The article was concluded by illustrative examples.

As observed in 111, 91, Kummer confluent hypergeometric functions appear naturally when considering the MID property for time-delay systems. In 75, we have established new properties of Kummer functions with integer coefficients, showing a new representation formula for these functions in terms of iterated integrals of an exponential kernel.

It has been known since at least 113 that, in many examples, spectral values of large multiplicity of a time-delay system appear naturally when minimizing the spectral abscissa of the system, but a general analysis of this phenomenon is still missing from the literature. The article 42 has considered, in its first part, the case of second-order retarded delay-differential equations with first-order delayed term coming from a full state feedback, proving that, in this configuration, triple real roots or pairs of complex-conjugate roots of multiplicity 2 each are minimizers of the spectral abscissa. The proofs rely on perturbation theory of nonlinear eigenvalue problems and exploit the quasi-convexity of the spectral abscissa function. In the second part, 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 translate into restrictions on the tunable parameters in the closed-loop quasi-polynomial.

6.4 New developments on the coexistence of real roots inducing dominancy

Participants: Souad Amrane [Université Mouloud Mammerie Tizi-Ouzou, Algérie], Fazia Bedouhene [Université Mouloud Mammerie Tizi-Ouzou, Algérie], Islam Boussaada [DISCO], Silviu-Iulian Niculescu [DISCO], Timothée Schmoderer [DISCO], Cyprien Tamekue [Laboratoire des Signaux et Systèmes (L2S)].

The spectral property coexistent-real-roots-inducing-dominancy (CRRID) first introduced in 82, 88 has the originality of assigning a maximal number of simple real zeros of the characteristic function rather than a multiple zero as is proposed in the MID approach. Thus, compared to the MID, the CRRID allows more parametric freedom and in general more robust design. During the past year, several new developments on the CRRID property were established in the team.

We contributed in 46 by showing that the coexistence of the maximal number of real spectral values of generic single-delay retarded second-order differential equations guarantees the realness of the rightmost spectral value. From a control theory standpoint, this entails that a delayed proportional-derivative (PD) controller can stabilize a delayed second-order differential equation. By assigning the maximum number of negative roots to the corresponding characteristic function (a quasipolynomial), we establish the conditions for asymptotic stability. If the assigned real spectral values are uniformly distributed, we specify a necessary and sufficient condition for the rightmost root to be negative, thus guaranteeing the exponential decay rate of the systems solutions. We illustrate the proposed design methodology in the delayed PD control of the damped harmonic oscillator.

As emphasized in 50, it appears that the CRRID is an appropriate attack angle to investigate the MID property in low multiplicities. Indeed, despite the fact that this assumption is not met, an analytical proof for the MID is proposed. The coexistence of real spectral values is the main ingredient. The obtained result is illustrated through the stabilization of an unstable second-order plant by a delayed PD controller.

Finally, in 79, a control-oriented delay-based modeling of a one-layer neural network of Hopfield-type subject to an external input designed as delayed feedback is investigated. The specificity of such a model is that it makes the considered neuron less susceptible to seizure caused by its inherent dynamic instability. This modeling exploits a recently set partial pole placement for linear functional differential equations, which relies on the coexistence of real spectral values, allowing the explicit prescription of the closed-loop solution's exponential decay. The proposed framework improves some pioneering and scarce results from the literature on the characterization of the exact solution's exponential decay when a simple real spectral value exists. Indeed, it improves neural stability when the inherent dynamic is stable and provides insights into the design of a one-layer neural network that can be stabilized exponentially with delayed feedback and with a prescribed decay rate regardless of whether the inherent neuron dynamic is stable or unstable.

6.5 Implementation of Stabilizing Controllers for Retarded Delay Systems

Participants: Catherine Bonnet [DISCO], Suat Gumussoy [Siemens Technology Princeton], Hitay Özbay [Bilkent University], Mustafa Oguz Yegin [Bilkent University].

We have proposed in 51 a way to efficiently implement Hinfty-stabilizing controllers for delay systems of retarded type. We indeed chose a particular controller (central controller) in the set of all stabilizing controllers and decomposed its transfer function in terms of stable delay systems and Finite Impulse Response (FIR) Systems. The central controller is then written in such a way that the corresponding Matlab/Simulink implementation scheme is obvious. We illustrated that all stabilizing controllers can be implemented using stable components only around the central controller. We also discussed stability robustness in the presence of implementation errors. The method was illustrated in an example.

6.6 Stability analysis of systems with delay-dependent coefficients and commensurate delays

Participants: Islam Boussaada [DISCO], Chi Jin [TuSimple Inc. Shanghai,China], Keqin Gu [Illinois University Edwardsville, USA], Qin Ma [Nanjing University of Science and Technology, China], Silviu-Iulian Niculescu [DISCO].

Systems with coefficients depending on time-delay may arise from various scientific disciplines, such as population models with age structure, the stellar dynamos, and hematopoiesis dynamics. In the SISO case, this class of system has been the subject of a thesis of Jin Chi (DISCO 2015-2017) where an analysis method has been proposed, see for instance 108. A generalizion to the MIMO case is obtained in 32 where a method of stability analysis of linear time-delay systems with commensurate delays and delay-dependent coefficients is proposed. 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 OrlandoÕs 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. An example is presented to illustrate how a stability analysis can be conducted if some non-degeneracy assumptions are violated.

6.7 Spectral methods for the stability of linear hyperbolic systems

Participants: Kais Ammari [Université de Monastir, Tunisie], Jean Auriol [Laboratoire des Signaux et Systèmes (L2S)], Ismaïla Balogoun [DISCO], Islam Boussaada [DISCO], Guilherme Mazanti [DISCO], Silviu Niculescu [DISCO], Timothée Schmoderer [DISCO], Sami Tliba [Laboratoire des Signaux et Systèmes (L2S)].

Systems of first-order hyperbolic partial differential equations (PDEs) have been extensively studied over the years due to their application in modeling various physical phenomena, such as drilling devices, water management systems, aeronomy, cable vibration dynamics, pipelines, and traffic networks 87, 106, 83, 100. When there are no in-domain coupling terms, the method of characteristics can be used to relate the behavior of these systems to time-delay systems, a link extensively explored in the literature 30, 96, 97, 98, 118. The situation is more delicate when in-domain coupling terms are present, but, as recently shown in 84, 90, a transformation into a time-delay system is still possible, using first a backstepping transformation to convert the in-domain couplings into integral boundary terms, and then applying the method of characteristics to obtain a time-delay system, which presents a distributed delay.

In 74, we explore the latter transformation in order to establish a necessary and sufficient stability condition for a class of two coupled first-order linear hyperbolic partial differential equations with in-domain coupling. The stability of the associated time-delay system, which is of neutral type, is studied through spectral methods, using Cauchy's argument principle in order to count the number of unstable roots. This is done by adapting to the present setting the Stépán–Hassard argument variation approach for retarded time-delay systems, first introduced in 119 and then refined in 105. Our theoretical findings are also validated in 74 through simulations.

In 46 the boundary control of the transport equation is addressed. Namely, based on the CRRID property, a control method is proposed, which is merely a delayed output feedback relying on a partial pole placement idea, that consists in assigning an appropriate exponential decay rate to the closed-loop system's solution. The proposed control structure appearing in the transport boundary, which has proven its effectiveness in controlling finite dimensional systems, consists of an autoregressive relation linking the transport equation's input and output. The obtained result provides an analytical lower bound for the solution's exponential decay.

As a mechanical engineering application, the transverse vibration midpoint control of a string is considered in 61. More precisely, we explore the use of a control block that includes delays in both input and output signals under the assumption that the memory-window length is also a parameter of the control law. Relatively simple to construct, such a control law was successfully used in the boundary control of the transport and wave equations. The effectiveness of the proposed method is illustrated through numerical simulations and the corresponding finite-difference scheme is detailed.

In 48, we design a controller for an interconnected system consisting of a linear Stochastic Differential Equation (SDE) actuated through a linear hyperbolic Partial Differential Equation (PDE). Our approach aims to minimize the variance of the state of the SDE component. We leverage a backstepping technique to transform the original PDE into an uncoupled stochastic PDE. As such, we reformulate our initial problem as the control of a delayed SDE with a non-deterministic drift. Under standard controllability assumptions, we design a controller steering the mean of the states to zero while keeping its covariance bounded. As final step, we address the optimal control of the delayed SDE employing Artsteins transformation and Linear Quadratic stochastic control techniques.

6.8 Averaging for time-varying systems

Participants: Rami Katz [Tel Aviv Univ], Frédéric Mazenc, Emilia Fridman [Tel Aviv Univ].

We have have developed a new avaraging technique for fast-varying continuous-time systems.

In 33, we treated the input-to-state stability-like estimates for perturbed linear continuous-time systems with multiple time-scales, under the assumption that the averaged unperturbed system is exponentially stable. Such systems contain rapidly-varying, piecewise continuous and almost periodic coefficients with small parameters. Our method relies on a novel delay-free system transformation in conjunction with a new system presentation, where the rapidly-varying coefficients are scalars that have zero average. We employ time-varying Lyapunov functions for ISS-like analysis. The analysis yields LMI conditions, leading to explicit bounds on the small parameters, decay rate and ISS gains. The novel system presentation plays a crucial role in the ISS-like analysis by allowing to derive essentially less conservative upper bounds on terms containing the small parameters. We further extended our approach to rapidly-varying systems subject to either discrete (constant or fast-varying) or distributed delays, where our approach decouples the effects of the delay and small parameters on the stability of the system and leads to LMI conditions for stability of systems with non-small delays. Numerical examples showed that, compared to the existing results, the approach essentially enlarges the small parameter and delay bounds for which the ISS-like/stability property of the original system is preserved.

6.9 Local Halanay's Inequality

Participants: Frédéric Mazenc, Michael Malisoff [LSU].

In contributions 35 and 56, we provided a local version of an approach to proving asymptotic stability that is based on the celebrate Halanay’s inequality. Our results are applicable to a family of nonlinear systems that contain state and input delays. We determine 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.

6.10 Extremum seeking

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

We solved in 55 a gradient based bounded extremum seeking problem for single-variable static maps in the presence of time-varying piecewise continuous measurement uncertainty. Instead of using previously reported averaging-based methods, we introduced a new state transformation, allowing us to use new comparison function and generalized Lyapunov function approaches to obtain our ultimate bounds on the parameter estimation error. We illustrated significant advantages of our new method, including less restrictive conditions on the extremum seeking parameters, as compared with previous methods.

6.11 Stability Analysis of Piecewise affine systems

Participants: Giorgio Valmorbida, Joao Manoel Gomes da Silva Jr [Universidade Federal do Rio Grande do Sul], Leonardo Cabral [Universidade Federal do Rio Grande do Sul], Diego Munoz Carpintero [Universidad O'Higgins, Chile].

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 29. 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.

We have also studied continuous-time systems with piecewise quadratic Lyapunov functions 28. The proposed stabilty conditions allowed to compute tight convergence rates with refined partition and more complex tests for copositivity.

For linear systems with state constraints, we proposed a parametrization of piecewise-quadratic Lyapunov functions to compute compute nonlinear control laws allowing to enlarge invariant sets 43. The main advantage of the proposed method is the computation of asymmetric invariant sets with low memory footprint.

6.12 Implicit Neural Networks and Piecewise affine functions from Model Predictive Control

Participants: Giorgio Valmorbida, Ross Drummond [University of Sheffield].

Model predictive control (MPC) for linear systems with quadratic costs and linear constraints were shown to admit an exact representation as an implicit neural network. A method to “unravel” the implicit neural network of MPC into an explicit one is also proposed. As well as building links between model-based and data-driven control, these results emphasize the capability of implicit neural networks for representing solutions of optimisation problems, as such problems are themselves implicitly defined functions.

6.13 Nonsmooth optimization algorithms through duality

Participants: Guilherme Mazanti [DISCO], Thibault Moquet [DISCO], Laurent Pfeiffer [DISCO].

The Frank–Wolfe Algorithm (FWA), also known as Conditional Gradient Algorithm, is an iterative minimization algorithm for minimizing a convex function with Lipschitz continuous gradient over a closed convex bounded subset $K$ of a Banach space. Introduced in 101, this method assumes that one can easily minimize linear functions orver $K$, through a so-called Linear Minimization Oracle (LMO), and proceeds by using the result of this oracle to update the candidate to optimality through convex combinations. Many extensions of the FWA were considered in the literature, such as those from 102, 117, 120.

In 78, we have proposed an extension of the FWA, named Dualized Level-Set (DLS) algorithm, which 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 dual of a cutting-plane algorithm, a fact previously observed and exploited in 85, 86, 121. The DLS algorithm essentially results from a dualization of a specific cutting-plane algorithm from 110, based on projections on some level sets, and it generates a sequence of primal-dual candidates. We prove that the corresponding sequence of primal-dual gaps converges to 0 with a rate of $O(1/\sqrt{t})$. With respect to other extensions of the FWA to nonsmooth costs, the main novelty here is that we do not rely on regularizations of the nonsmooth term, dealing with it through duality instead.

6.14 Existence and asymptotic behavior of equilibria for general optimal-exit mean field games

Participants: Guilherme Mazanti [DISCO].

In recent years, many works, such as 95, 89, 103, 104, 94, have considered mean field games (MFGs) in which the time at which an agent exits the game is part of their optimization criterion. In particular, 112, 99, 114, 115, 116 have considered several variants of a similar MFG model motivated by crowd motion in which agents evolve in a given set and wish to reach a given target set, typically in minimal time and possibly with an additional cost in the exit position. In these MFGs, 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, an interaction that may model, in particular, congestion phenomena.

The main aim of 36 was to provide a unifying theory for the existence of equilibria of this class of MFGs, and to detail results on the asymptotic behavior of equilibria and on their dependence with respect to the initial distribution of agents. The main result of 36 on existence of equilibria generalizes all previous existence results proved in 112, 99, 114, 115, 116 and can also be applied to other classes of games, such as MFGs on networks. In addition, 36 also provides results on the convergence of the distribution of agents to a limit distribution, giving precise convergence rates. Finally, it is shown in 36 that, even though equilibria and limit distributions of such MFGs may not be continuous functions of the initial distribution of agents $m_0$, they are necessarily upper semicontinuous set-valued functions of $m_0$, which is shown to be the best result one may expect in full generality.

6.15 Mechanical engineering application

Participants: Islam Boussaada [DISCO], Jaroslav Busek [Czech Technical University in Prague, Czech Republic], Matej Kure [Czech Technical University in Prague, Czech Republic], Silviu Niculescu [DISCO], Tomas Vyhlidal [Czech Technical University in Prague, Czech Republic], Wim Michiels [KU Leuven, Belgium].

In 34, a combination of analytical and numerical time-delay-system spectrum-shaping tools are applied to the design of the robust delayed resonator with an acceleration feedback. First, the delayed resonator model is turned into a dimension-less form with the objective to generalize the derived results. The main theoretical result is then provided as a complete parameterization of the proposed resonator feedback with two delay terms to assign a pair of roots with multiplicity two on the imaginary axis. In the frequency-domain, the double roots are projected to widening the stop-band in the active absorber frequency response, which increases its robustness in vibration suppression. On the other hand, they have a destabilizing effect on the overall system dynamics. The stabilization is subsequently performed by an additional controller via spectral optimization. The design is thoroughly validated by both simulations and experiments where the results are compared with the classical delayed resonator with lumped delay acceleration feedback.

6.16 Chemostat with delay

Participants: Frédéric Mazenc, Gonzalo Robeldo [Univ. of Chile], Daniel Sepulveda [UTEM].

In 39, we have revisited a recently introduced chemostat model of one–species with a periodic input of a single nutrient which is described by a system of delay differential equations. Previous results provided sufficient conditions ensuring the existence and uniqueness of a periodic solution for arbitrarily small delays. This paper partially extends these results by proving—with the construction of Lyapunov–like functions—that the evoked periodic solution is globally asymptotically stable when considering Monod uptake functions and a particular family of nutrient inputs.

6.17 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 their individual profit, calculated as the revenue from selling fish minus the cost of moving their 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.

The paper 109 has been published in Nonlinearity in 2024. In this paper, we assume that, when no fisherman is present, the fishes' population is invading (mathematically, there is an invading travelling front). We prove that fishermen, when acting selfishly, each in their own best interest, might lead to a reversal of the travelling wave and, consequently, to an extinction of the global population. We then show that, at least in some cases, if the fishermen coordinated instead of acting selfishly, each of them could make more benefit, while still guaranteeing the survival of the population.

In 76, 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 109. 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.

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

10.1 Major publications

• 1 inproceedingsA. L.Ariádne L.J. Bertolin, R. C.Ricardo C.L.F. Oliveira, G.Giorgio Valmorbida and P. L.Pedro L.D. Peres. Output feedback control of discrete-time Lur'e systems through FIR Zames-Falb multipliers.IFAC World Congress 2023562Yokohama, France2023, 9528-9533HALDOI
• 2 inproceedingsA. L.Ariádne L J Bertolin, P. L.Pedro L D Peres, R. C.Ricardo C L F Oliveira and G.Giorgio Valmorbida. An LMI-based iterative algorithm for state and output feedback stabilization of discrete-time Lur'e systems.2020 59th IEEE Conference on Decision and Control (CDC)Jeju Island, South KoreaIEEEDecember 2020, 2561-2566HALDOI
• 3 articleC.Catherine Bonnet, A. R.André R. Fioravanti and J. R.Jonathan R. Partington. Stability of Neutral Systems with Commensurate Delays and Poles Asymptotic to the Imaginary Axis.SIAM Journal on Control and Optimization492March 2011, 498-516HAL
• 4 articleC.C. Bonnet and J.J.R. Partington. Stabilization of some fractional delay systems of neutral type.Automatica432007, 2047--2053
• 5 articleI.Islam Boussaada, I.-C.Irinel-Constantin Morarescu and S.-I.Silviu-Iulian Niculescu. Inverted pendulum stabilization: characterization of codimension-three triple zero bifurcation via multiple delayed proportional gains.Systems Control Lett.822015, 1--9
• 6 inproceedingsC.Cyrille Chenavier, R.Rosane Ushirobira and G.Giorgio Valmorbida. A geometric stabilization of planar switched systems.IFAC 2020 - 21st IFAC World CongressBerlin, GermanyJuly 2020HAL
• 7 inproceedingsA.Ali Diab, W.William Pasillas-Lépine and G.Giorgio Valmorbida. A Steer-by-Wire Control Architecture Robust to High Assistance Gains and Large Transmission Delays.Proceeding of the 15th International Symposium on Advanced Vehicle ControlThe 15th International Symposium on Advanced Vehicle ControlProceeding of the 15th International Symposium on Advanced Vehicle ControlKanagawa, JapanSociety of Automotive Engineers of JapanSeptember 2022HAL
• 8 articleA.Ali Diab, G.Giorgio Valmorbida and W.William Pasillas-Lépine. Linear assistance filter design for electric power steering systems with improved delay margin.International Journal of ControlApril 2023, 1 - 18HALDOI
• 9 inproceedingsM.Mathias Giordani Titton, J. M.João Manoel Gomes da Silva Jr. and G.Giorgio Valmorbida. Stability of Sampled-Data Control for Lurie Systems with Slope-Restricted Nonlinearities.Congresso Brasileiro de Automática - 2020Porto Alegre, BrazilsbabraNovember 2020HALDOI
• 10 inproceedingsL. B.Leonardo B Groff, J. M.João Manoel Gomes Da Silva and G.Giórgio Valmórbida. Regional Stability of Discrete-Time Linear Systems Subject to Asymmetric Input Saturation.CDC 2019 - 58th IEEE Conference on Decision and ControlNice, FranceDecember 2019HAL
• 11 inproceedingsL. B.Leonardo B Groff, G.Giórgio Valmórbida and J. M.João M Gomes Da. Stability analysis of piecewise affine discrete-time systems.CDC 2019 - 58th IEEE Conference on Decision and ControlNice, FranceDecember 2019HAL
• 12 inproceedingsM.Morten Hovd and G.Giorgio Valmorbida. Solving LP-MPC problems using ramp functions*.2023 IEEE Conference on Control Technology and Applications (CCTA)Bridgetown, FranceIEEEAugust 2023, 445-450HALDOI
• 13 bookM.M. Malisoff and F.F. Mazenc. Constructions of Strict Lyapunov Functions.Communications and Control Engineering SeriesSpringer-Verlag London Ltd.2009
• 14 articleF.F. Mazenc, M.M. Malisoff and S.-I.S.-I. Niculescu. Reduction Model Approach for Linear Time-Varying Systems with Delays.IEEE Transactions on Automatic Control5982014, 2068--2014
• 15 bookW.Wim. Michiels and S.-I.Silviu-Iulian. Niculescu. S.-I.Silviu-Iulian Niculescu and W.Wim Michiels, eds. Stability, Control, and Computation for Time-Delay Systems.Philadelphia, PASociety for Industrial and Applied Mathematics2014
• 16 bookS.-I.S.-I. Niculescu. Delay Effects on Stability: a Robust Control Approach.269Lecture Notes in Control and Information SciencesSpringer2001
• 17 articleI.Isabelle Queinnec, S.Sophie Tarbouriech, G.Giorgio Valmorbida and L.Luca Zaccarian. Design of Saturating State-Feedback with Sign-Indefinite Quadratic Forms.IEEE Transactions on Automatic Control2021HALDOI
• 18 bookM. B.Martha Belem Saldivar, I.Islam Boussaada, H.Hugues Mounier and S.-I.Silviu-Iulian Niculescu. Analysis and Control of Oilwell Drilling Vibrations.Springer International PublishingMay 2015HAL
• 19 articleM. G.Mathias Giordani Titton, J. M.João Manoel Gomes da Silva Jr., G.Giorgio Valmorbida and M.Marc Jungers. Stability analysis of Lure systems under aperiodic sampled‐data control.International Journal of Robust and Nonlinear Control3312August 2023, 7130-7153HALDOI
• 20 articleG.G. Valmorbida, M.M. Ahmadi and A.A. Papachristodoulou. Stability Analysis for a Class of Partial Differential Equations via Semidefinite Programming.IEEE Transactions on Automatic Control616June 2016, 1649-1654
• 21 inproceedingsG.Giorgio Valmorbida and F.Francesco Ferrante. On Quantization in Discrete-Time Control Systems: Stability Analysis of Ternary Controllers.CDC 2020 - 59th IEEE Conference on Decision and ControlIEEEJeju Island (virtual), South KoreaDecember 2020HAL
• 22 articleG.G. Valmorbida, S.S. Tarbouriech and G.G. Garcia. Design of Polynomial Control Laws for Polynomial Systems Subject to Actuator Saturation.IEEE Transactions on Automatic Control587July 2013, 1758-1770

10.2 Publications of the year

10.3 Cited publications

• 82 articleS.Souad Amrane, F.Fazia Bedouhene, I.Islam Boussaada and S.-I.Silviu-Iulian Niculescu. On Qualitative Properties of Low-Degree Quasipolynomials: Further remarks on the spectral abscissa and rightmost-roots assignment.Bulletin mathématique de la Société des Sciences mathématiques de Roumanie612018, 361 - 381HALback to text
• 83 articleJ.J. Auriol, I.I. Boussaada, R. J.R. J. Shor, H.H. Mounier and S.-I.S.-I. Niculescu. Comparing advanced control strategies to eliminate stick-slip oscillations in drillstrings.IEEE Access102022, 10949--10969back to text
• 84 articleJ.J Auriol and F.F Di Meglio. An explicit mapping from linear first order hyperbolic PDEs to difference systems.Systems & Control Letters1232019, 144--150back to text
• 85 articleF.Francis Bach. Duality between subgradient and conditional gradient methods.SIAM J. Optim.2512015, 115--129URL: https://doi.org/10.1137/130941961DOIback to text
• 86 articleF.Francis Bach. Learning with Submodular Functions: A Convex Optimization Perspective.Foundations and Trends in Machine Learning62--32013, 145--373URL: http://dx.doi.org/10.1561/2200000039DOIback to text
• 87 bookG.Georges Bastin and J.-M.Jean-Michel Coron. Stability and boundary stabilization of 1-D hyperbolic systems.88Progress in Nonlinear Differential Equations and their ApplicationsSubseries in ControlBirkhäuser/Springer, [Cham]2016, xiv+307URL: https://doi.org/10.1007/978-3-319-32062-5DOIback to text
• 88 articleF.Fazia Bedouhene, I.Islam Boussaada and S.-I.Silviu-Iulian Niculescu. Real spectral values coexistence and their effect on the stability of time-delay systems: Vandermonde matrices and exponential decay.Comptes Rendus. Mathématique3589-10September 2020, 1011-1032HALDOIback to text
• 89 articleC.Charles Bertucci. Optimal stopping in mean field games, an obstacle problem approach.J. Math. Pures Appl. (9)1202018, 165--194DOIback to text
• 90 articleD.D. Bou Saba, F.F. Bribiesca-Argomedo, J.J. Auriol, M.M. Di Loreto and F.F. Di Meglio. Stability Analysis for a Class of Linear $22$ Hyperbolic PDEs Using a Backstepping Transform.IEEE Transactions on Automatic Control6572019, 2941--2956back to text
• 91 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.C. R. Math. Acad. Sci. Paris3602022, 349--369DOIback to textback to text
• 92 articleI.Islam Boussaada and S.-I.Silviu-Iulian Niculescu. Characterizing the codimension of zero singularities for time-delay systems: a link with Vandermonde and Birkhoff incidence matrices.Acta Appl. Math.1452016, 47--88URL: https://doi-org.ezproxy.universite-paris-saclay.fr/10.1007/s10440-016-0050-9DOIback to text
• 93 inproceedingsI.Islam Boussaada, H. U.Hakki Ulaş Unal and S.-I.Silviu-Iulian Niculescu. Multiplicity and stable varieties of time-delay systems: A missing link.22nd International Symposium on Mathematical Theory of Networks and Systems (MTNS)2016back to text
• 94 articleL.Luciano Campi and M.Markus Fischer. $N$-player games and mean-field games with absorption.Ann. Appl. Probab.2842018, 2188--2242DOIback to text
• 95 articleR.René Carmona, F.François Delarue and D.Daniel Lacker. Mean field games of timing and models for bank runs.Appl. Math. Optim.7612017, 217--260DOIback to text
• 96 articleY.Yacine Chitour, S.Swann Marx and G.Guilherme Mazanti. One-dimensional wave equation with set-valued boundary damping: well-posedness, asymptotic stability, and decay rates.ESAIM Control Optim. Calc. Var272021, Paper No. 84, 62 pp.DOIback to text
• 97 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. Media1142016, 563--601URL: https://doi.org/10.3934/nhm.2016010DOIback to text
• 98 articleK. L.K. L Cooke and D. W.D. W Krumme. Differential-difference equations and nonlinear initial-boundary value problems for linear hyperbolic partial differential equations.Journal of Mathematical Analysis and Applications2421968, 372--387back to text
• 99 articleS.Samer Dweik and G.Guilherme Mazanti. Sharp semi-concavity in a non-autonomous control problem and $L^p$ estimates in an optimal-exit MFG.NoDEA Nonlinear Differential Equations Appl.2722020, Paper No. 11, 59 pp.DOIback to textback to text
• 100 articleN.N. Espitia, J.J. Auriol, H.H. Yu and M.M. Krstic. Traffic flow control on cascaded roads by event-triggered output feedback.International Journal of Robust and Nonlinear Control32102022, 5919--5949back to text
• 101 articleM.Marguerite Frank and P.Philip Wolfe. An algorithm for quadratic programming.Naval Research Logistics Quarterly31-21956, 95-110URL: https://doi.org/10.1002/nav.3800030109DOIback to text
• 102 inproceedingsG.Gauthier Gidel, F.Fabian Pedregosa and S.Simon Lacoste-Julien. Frank-Wolfe splitting via augmented Lagrangian method.International Conference on Artificial Intelligence and StatisticsPMLR2018, 1456--1465back to text
• 103 articleD. A.Diogo A. Gomes and S.Stefania Patrizi. Obstacle mean-field game problem.Interfaces Free Bound.1712015, 55--68DOIback to text
• 104 articleP. J.Philip Jameson Graber and C.Charafeddine Mouzouni. On mean field games models for exhaustible commodities trade.ESAIM Control Optim. Calc. Var.262020, Paper No. 11, 38URL: https://doi.org/10.1051/cocv/2019008DOIback to text
• 105 articleB.BD Hassard. Counting roots of the characteristic equation for linear delay-differential systems.Journal of Differential Equations13621997, 222--235back to text
• 106 articleA.A. Hayat. Boundary stabilization of 1D hyperbolic systems.Annual Reviews in Control522021, 222--242back to text
• 107 articleE.Einar Hille. Oscillation theorems in the complex domain.Trans. Amer. Math. Soc.2341922, 350--385URL: https://doi.org/10.2307/1988884DOIback to text
• 108 articleC.Chi Jin, K.Keqin Gu, I.Islam Boussaada and S.-I.Silviu-Iulian Niculescu. Stability Analysis of a More General Class of Systems with Delay-Dependent Coefficients.IEEE Transactions on Automatic Control14January 2019, 1HALDOIback to text
• 109 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.Nonlinearity37112024, 115010back to textback to text
• 110 articleC.Claude Lemaréchal, A.Arkadii Nemirovskii and Y.Yurii Nesterov. New variants of bundle methods.Mathematical programming691995, 111--147back to text
• 111 articleG.Guilherme Mazanti, I.Islam Boussaada and S.-I.Silviu-Iulian Niculescu. Multiplicity-induced-dominancy for delay-differential equations of retarded type.J. Differential Equations2862021, 84--118DOIback to textback to text
• 112 articleG.Guilherme Mazanti and F.Filippo Santambrogio. Minimal-time mean field games.Math. Models Methods Appl. Sci.2982019, 1413--1464URL: https://doi.org/10.1142/S0218202519500258DOIback to textback to text
• 113 bookE.Edmund Pinney. Ordinary difference-differential equations.University of California Press, Berkeley-Los Angeles1958, xii+262back to text
• 114 articleS.Saeed Sadeghi Arjmand and G.Guilherme Mazanti. Multipopulation minimal-time mean field games.SIAM J. Control Optim.6042022, 1942--1969DOIback to textback to text
• 115 articleS.Saeed Sadeghi Arjmand and G.Guilherme Mazanti. Nonsmooth mean field games with state constraints.ESAIM Control Optim. Calc. Var.282022, Paper No. 74, 42 pp.DOIback to textback to text
• 116 inproceedingsS.Saeed Sadeghi Arjmand and G.Guilherme Mazanti. On the characterization of equilibria of nonsmooth minimal-time mean field games with state constraints.2021 60th IEEE Conference on Decision and Control (CDC)2021, 5300--5305DOIback to textback to text
• 117 articleA.Antonio Silveti-Falls, C.Cesare Molinari and J.Jalal Fadili. Generalized Conditional Gradient with Augmented Lagrangian for Composite Minimization.SIAM Journal on Optimization304January 2020, 2687--2725URL: http://dx.doi.org/10.1137/19M1240460DOIback to text
• 118 articleM.M Slemrod. Nonexistence of oscillations in a nonlinear distributed network.Journal of Mathematical Analysis and Applications3611971, 22--40back to text
• 119 bookG.G. Stépán. Retarded Dynamical Systems: Stability and Characteristic Functions.Pitman research notes in mathematics seriesLongman Scientific & Technical1989back to text
• 120 inproceedingsA.Alp Yurtsever, O.Olivier Fercoq and V.Volkan Cevher. A conditional-gradient-based augmented Lagrangian framework.International Conference on Machine LearningPMLR2019, 7272--7281back to text
• 121 articleS.Song Zhou, S.Swati Gupta and M.Madeleine Udell. Limited memory Kelley's method converges for composite convex and submodular objectives.Advances in Neural Information Processing Systems312018back to text