Sparse approximation under non-negativity constraints naturally arises in several applications. Many sparse solvers can be directly extended to the non-negative setting. It is not the case of Orthogonal Matching Pursuit (OMP), a well-known sparse solver, which gradually updates the sparse solution support by selecting a new dictionary atom at each iteration. When dealing with non-negative constraints, the orthogonal projection computed at each OMP iteration is replaced by a non-negative least-squares (NNLS) subproblem whose solution is not explicit. Therefore, the usual recursive (fast) implementations of OMP do not apply. A Non-negative version of OMP (NNOMP) was proposed in the recent literature together with several variations. In my talk, I will first recall the principle of greedy algorithms, in particular NNOMP, and then, I will introduce our proposed improvements, based on the use of the active-set algorithm to address the NNLS subproblems. The structure of the active-set algorithm is indeed intrisically greedy. Moreover, the active-set algorithm can be called with a warm start, allowing us to fastly solve the NNLS subproblems. (Joint work with Charles Soussen (L2S), Jérôme Idier (LS2N), and El-Hadi Djermoune (CRAN).)

### S³ seminar : Non-negative orthogonal greedy algorithms for sparse approximation

**Thanh NGUYEN (CRAN, L2S)**

### Séminaire d'Automatique du Plateau de Saclay : Necessary and sufficient condition for exponential synchronization of nonlinear systems

**Vincent Andrieu (CNRS Researcher, LAGEP-CNRS, Université de Lyon 1, France)**

Based on recent works on transverse exponential stability, some necessary and sufficient conditions for the existence of a (locally) exponential synchronizer are established. We show that the existence of a structured synchronizer is equivalent to the existence of a stabilizer for the individual linearized systems (on the synchronization manifold) by a linear state feedback. This, in turns, is also equivalent to the existence of a symmetric covariant tensor field which satisfies a kind of Lyapunov inequality. Based on this property, we provide the construction of such synchronizer. We discuss then the possibility to achieve global synchronization.

Bio. Vincent Andrieu graduated in applied mathematics from “INSA de Rouen”, France, in 2001. After working in ONERA (French aerospace research company), he obtained a PhD degree from “Ecole des Mines de Paris” in 2005. In 2006, he had a research appointment at the Control and Power Group, Dept. EEE, Imperial College London. In 2008, he joined the CNRS-LAAS lab in Toulouse, France, as a “CNRS-chargé de recherche”. Since 2010, he has been working in LAGEP-CNRS, Université de Lyon 1, France. In 2014, he joined the functional analysis group from Bergische Universitäte Wuppertal in Germany, for two sabbatical years. His main research interests are in the feedback stabilization of controlled dynamical nonlinear systems and state estimation problems. He is also interested in practical application of these theoretical problems, and especially in the field of aeronautics and chemical engineering.

### Séminaire d'Automatique du Plateau de Saclay : Observer design for nonlinear systems

**Pauline Bernard (PhD, PSL Reserch University, Systems and Control Center, MINES ParisTech)**

Unlike for linear systems, no systematic method exists for the design of observers for nonlinear systems. However, observer design may be more or less straightforward depending on the coordinates we choose to express the system dynamics. In particular, some specific structures, called normal forms, have been identified for allowing a direct and easier observer construction. It follows that a common way of addressing the problem consists in looking for a reversible change of coordinates transforming the expression of the system dynamics into one of those normal forms, design an observer in those coordinates, and finally deduce an estimate of the system state in the initial coordinates via inversion of the transformation. This talk gives contributions to each of those three steps.

First, we show the interest of a new triangular normal form with continuous (non-Lipschitz) nonlinearities. Indeed, we have noticed that systems which are observable for any input but with an order of differential observability larger than the system dimension, may not be transformable into the standard Lipschitz triangular form, but rather into an "only continuous" triangular form. In this case, the famous high gain observer no longer is sufficient, and we propose to use homogeneous observers instead.

Another canonical form of interest is the Hurwitz linear form which admits a trivial observer. The question of transforming a nonlinear system into such a form has only been addressed for autonomous systems with the so-called Lunberger or Kazantzis-Kravaris observers. This design consists in solving a PDE and we show here how it can be extended to time-varying/controlled systems.

As for the inversion of the transformation, this step is far from trivial in practice, in particular when the domain and image spaces have different dimensions. When no explicit expression for a global inverse is available, numerical inversion usually relies on the resolution of a minimization problem with a heavy computational cost. That is why we have developed a method to avoid the explicit inversion of the transformation by bringing the observer dynamics (expressed in the canonical form coordinates) back into the initial system coordinates. This is done by dynamic extension, i.e. by adding some new coordinates to the system and transforming an injective immersion into a surjective diffeomorphism.

Bio. Pauline Bernard graduated from MINES ParisTech in 2014 with a Master degree in Applied Mathematics and Automatic Control. In 2017, she obtained her Ph.D. in Mathematics and Automatic Control at PSL Reserch University, prepared at the Systems and Control Center, MINES ParisTech under the supervision of Laurent Praly and Vincent Andrieu.

### Séminaire d’Automatique du plateau de Saclay : Stability analysis of discrete-time infinite-horizon control with discounted cost.

**Romain Postoyan (CNRS researcher, Centre de Recherche en Automatique de Nancy)**

We analyse the stability of general nonlinear discrete-time systems controlled by an optimal sequence of inputs that minimizes an inﬁnite-horizon discounted cost. First, assumptions related to the controllability of the system and its detectability with respect to the stage cost are made. Uniform semiglobal and practical stability of the closed-loop system is then established, where the adjustable parameter is the discount factor. Stronger stability properties are thereupon guaranteed by gradually strengthening the assumptions. Next, we show that the Lyapunov function used to prove stability is continuous under additional conditions, implying that stability has a certain amount of nominal robustness. The presented approach is ﬂexible and we show that robust stability can still be guaranteed when the sequence of inputs applied to the system is no longer optimal but near-optimal. We also analyse stability for cost functions in which the importance of the stage cost increases with time, opposite to discounting. Finally, we exploit stability to derive new relationships between the optimal value functions of the discounted and undiscounted problems, when the latter is well-deﬁned.

This is a joint work with Lucian Busoniu (TU Cluj, Romania), D. Nesic (University of Melbourne, Australia) and J. Daafouz (CRAN, Université de Lorraine).

Bio. Romain Postoyan received the master degree (``diplôme d'ingénieur'') in Electrical and Control Engineering from ENSEEIHT (France) in 2005. He obtained the M.Sc. by Research in Control Theory & Application from Coventry University (United Kingdom) in 2006 and the Ph.D. in Control Theory from Université Paris-Sud (France) in 2009. In 2010, he was a research assistant at the University of Melbourne (Australia). Since 2011, he is a CNRS researcher at the Centre de Recherche en Automatique de Nancy (France). He serves as an Associate Editor at the Conference Editorial Board of the IEEE Control Systems Society and for the journals: Automatica, IEEE Control Systems Letters, and IMA Journal of Mathematical Control and Information.

### Séminaire d'Automatique du Plateau de Saclay : Message-passing computation of the harmonic influence in social networks

**Paolo Frasca (CNRS Researcher, NeCS team, GIPSA-lab, Grenoble, France).**

The harmonic influence is a measure of node influence in social networks that quantifies the ability of a leader node to alter the average opinion of the network, acting against an adversary field node. The definition of harmonic influence assumes linear interactions between the nodes described by an undirected weighted graph; its computation requires to solve, for every node, a discrete Dirichlet problem associated to a grounded Laplacian. In this talk, I will describe a message-passing distributed algorithm that concurrently computes the harmonic influence of all nodes and provide a convergence analysis for it. The algorithm converges asymptotically, under the only assumption of the interaction Laplacian being symmetric. However, the convergence value does not in general coincide with the harmonic influence: simulations show that when the network has a larger number of cycles, the algorithm becomes slower and less accurate, but nevertheless provides a useful approximation. Simulations also indicate that the symmetry condition is not necessary for convergence and that performance (both in terms of speed and asymptotical error) scales well in the number of nodes of the graph.

Bio. Paolo Frasca received the Ph.D. degree in Mathematics for Engineering Sciences from Politecnico di Torino, Torino, Italy, in 2009. Between 2008 and 2013, he has held research and visiting positions at the University of California, Santa Barbara (USA), at the IAC-CNR (Rome, Italy), at the University of Salerno (Italy), and at the Politecnico di Torino. From 2013 to 2016, he has been an Assistant Professor at the University of Twente in Enschede, the Netherlands. In October 2016 he joined the CNRS as Researcher: he is currently affiliated with GIPSA-lab in Grenoble, France.

His research interests are in the theory of network systems and cyber-physical systems, with applications to robotic, sensor, infrastructural, and social networks. On these topics, Dr. Frasca has (co)authored more than fifty journal and conference papers and has given invited talks at several international institutions and events, including the 2015 SICE International Symposium on Control Systems in Tokyo. He is a recipient of the 2013 SIAG/CST Best SICON Paper Prize. He has been a visiting professor at the LAAS, Toulouse, France in 2016 and at the University of Cagliari, Italy in 2017.

Dr. Frasca has served as Associate Editor of several international conferences, including IEEE CDC, ACC, ECC, MTNS, IFAC NecSys, and is currently serving as Associate Editor for the International Journal of Robust and Nonlinear Control, the Asian Journal of Control, and the IEEE Control Systems Letters.

### Séminaire d'Automatique du Plateau de Saclay : Distributed Abstractions for Multi-Agent Systems Based on Robust Multi-Agent Control

**Dimitris Boskos (Postdoctoral researcher, Department of Automatic Control, School of Electrical Engineering, Royal Institute of Technology (KTH), Stockholm, Sweden)**

High level task planning for multi-agent systems constitutes a research area which has gained an emerging attention during the last two decades. While the agents' coordination is in principle based on the design of continuous interaction protocols, the derivation of high level plans requires a discrete representation of their dynamic behavior, also called abstraction, in order to leverage algorithmic tools for the plan synthesis.

In this talk we discuss the derivation of such abstractions for agents with continuous dynamics, comprising of feedback interconnection terms and additive bounded inputs, which provide the ability for high level planning under the coupled constraints. These dynamics are also motivated by multi-agent coordination protocols which are robust with respect to the additional input part. We will present such a cooperative control framework, which guarantees that network connectivity is robustly maintained with respect to bounded additive inputs. Furthermore, a modification of the feedback design ensures forward invariance of the agents' trajectories inside a convex workspace, without affecting the inputs' robustness bounds.

In order to derive the agents' distributed symbolic models, we determine space-time discretizations which establish that each agent's abstraction has at least one outgoing transition from every discrete state. The symbolic model of each agent is based on the knowledge of its neighbors' discrete positions and the transitions are performed through hybrid control laws, which can drive the agent to its possible successor states. As an extension of these results we also consider a varying degree of decentralization and build each abstract model based on discrete information up to a tunable distance in the communication graph. Finally, we discuss the derivation of online abstractions, by discretizing over approximations of the agents' reachable sets over a bounded time horizon.

Bio. Dimitris Boskos was born in Athens, Greece in 1981. He has received the Diploma in Mechanical Engineering from the National Technical University of Athens (NTUA), Greece, in 2005, the M.Sc. in Applied Mathematics from the NTUA in 2008 and the Ph.D. in Applied mathematics from the NTUA in 2014. Since August 2014, he is a Postdoctoral Researcher at the Department of Automatic Control, School of Electrical Engineering, Royal Institute of Technology (KTH), Stockholm, Sweden. His research interests include distributed control of multi-agent systems, formal verification and observer design for nonlinear systems.

### Séminaire d'Automatique du Plateau de Saclay : Optimal control problems with oscillations, concentrations, and discontinuities.

**Didier Henrion (CNRS Senior Researcher, LAAS-CNRS & Professor, Faculty of Electrical Engineering, Czech Technical University)**

Optimal control problems with oscillation (chattering controls) and concentration (impulsive controls) can have integral performance criteria such that concentration of the control signal occurs at a discontinuity of the state signal. Techniques from functional analysis (extensions of DiPerna-Majda measures from the partial differential equations literature) are developed to give a precise meaning of the integral cost and to allow for the sound application of numerical methods. We show how this can be achieved for the Lasserre hierarchy of semidefinite programming relaxations. This includes in particular the use of compactification techniques allowing for unbounded time, state and control.

Bio. Didier Henrion is a CNRS Senior Researcher at LAAS, an engineering laboratory in Toulouse, France. He is also a Professor at the Faculty of Electrical Engineering at the Czech Technical University in Prague, Czechia. Since 1994 he has been developing constructive tools for addressing mathematical problems arising from systems control and optimization.

### Séminaire d'Automatique du plateau de Saclay : Stabilization of nonlinear infinite-dimensional systems subject to saturations

**Swann Marx (Postdoctoral researcher, LAAS-CNRS)**

This presentation provides contributions in stabilization methods for nonlinear dynamical systems. In particular, it focuses on the analysis of infinite-dimensional systems subject to saturated inputs.

In the first part, we will introduce a more general class of saturations than the one known for finite-dimensional systems. When bounding a linear stabilizing feedback law with such nonlinearity, a well-posedness result together with an attractivity result will be stated for systems whose open-loop is described by (possibly nonlinear) operators generating strongly continuous semigroup of contractions. The attractivity result will be proved by using the LaSalle's Invariance Principle together with some precompactness properties.

In the second part, a particular nonlinear partial differential equation is studied, namely the Korteweg-de Vries equation, that models long waves in water of relatively shallow depth. A control actuating on a small part of the channel will be considered. This control will be modified with two different types of saturations. The attractivity result will be proved by using Lyapunov argument and a contradiction argument. Finally, the results will be illustrated with some numerical simulations.

Bio. Swann Marx graduated in 2014 from "Ecole Supérieure de Cachan", France. He got his Ph.D. in the Departement of Automatic at the GIPSA-lab, in Grenoble, France. He is currently a postdoctoral researcher at the LAAS-CNRS, in Toulouse, France. His main research interests are stabilization of partial differential equations with constrained inputs, output feedback stabilization and optimal control of nonlinear partial differential equations.

### Inertia in inverter-dominated power networks.

**Pooya MONSHIZADEH (PhD student at University of Groningen, The Netherlands)**

Along with the emergence of the renewable energy sources in power networks, and consequently the increasing usage of power converters, new issues and concerns regarding stability of the grid have arisen. Recently, the problem of low inertia of inverter dominated systems has been extensively investigated. In this talk, I address the problem of stability and frequency regulation of a recently proposed inverter. In this type of inverter, the DC-side capacitor emulates the inertia of a synchronous generator. First, I discuss remodeling the dynamics from the electrical power perspective. Using this model, it can be shown that the system is stable if connected to a constant power load, and the frequency can be regulated by a suitable choice of the controller. I elaborate the analysis of the stability of a network of inverters with capacitive inertia, and show that frequency regulation can be achieved by using an

appropriate controller design.

### Séminaire d’Automatique du plateau de Saclay : Non-Markovian Quantum Feedback Networks

**John Gough (Institute of Mathematics and Physics, Aberystwyth University)**

We will recall the theory of Markovian Quantum feedback Networks, and explain some recent models with non-Markovian behaviour coming from physical requirements.

The concept of a controlled flow of a dynamical system, especially when the controlling process feeds information back about the system, is of central importance in control engineering, and we build on the ideas of by Bouten and van Handel to develop a general theory of quantum feedback. We elucidate the relationship between the controlling processes Z and the measured process Y, and to this end make a distinction between what we call the input picture and the output picture.

The theory is general enough to include a modulating filter which processes the measurement readout Y before returning to the system. This opens up the prospect of applying very general engineering feedback control techniques to open quantum systems in a systematic manner, and we consider a number of specific modulating filter problems.

Bio. John E. Gough was born in Drogheda, Ireland, in 1967. He received the B.Sc. and M.Sc. in Mathematical Sciences and the Ph.D. degree in Mathematical Physics from the National University of Ireland, Dublin, in 1987, 1988 and 1992 respectively. He was reader in Mathematical Physics at the Department of Mathematics and Computing, Nottingham-Trent University, up until 2007. He then joined the Institute of Mathematics and Physics at Aberystwyth University as established chair of Mathematics. He has held visiting positions at the University of Rome Tor Vergata, EPFL Lausanne, UC Santa Barbara and the Hong Kong Polytechnic University. His research interests include quantum probability, measurement and control of open quantum dynamical systems, and quantum feedback networks.

### Séminaire d'Automatique du Plateau de Saclay : Quantum reservoir engineering to control light with atoms.

**Alain Sarlette (QUANTIC Lab, INRIA Paris & SYSTeMS research group, Ghent University)**

This talk will give a simple introduction to a quantum stabilization technique called "reservoir engineering", which builds on the dissipation induced on a target system by its interaction with an open auxiliary system. We will introduce the technique, recall some older results obtained in our group and finally present our latest results in which we study the effect on the target system, of entanglement "in time" in the auxiliary system. This is joint work with my postdoc Zibo Miao.

Bio. Alain Sarlette has an engineering degree (applied physics) and a PhD (systems and control theory) from the University of Liège, Belgium. He has been a visiting researcher at Princeton University, Mines Paris-Tech, IIT Bombay among others. He is currently assistant professor at Ghent University (Belgium) and Senior Researcher (CR1) at INRIA Paris in the QUANTIC lab. His research interests include nonlinear and geometric control, coordination algorithms, and mainly algorithms and control for quantum technologies.

### Séminaire d'Automatique du plateau de Saclay : Some Mathematical and Numerical questions on the Lindblad equation. Application to Quantum Control.

**Claude Le Bris (Ecole des Ponts & Inria)**

We overview a series of joint works with P. Rouchon (Mines de Paris) and other collaborators devoted to the mathematical analysis and the numerical simulation of high-dimensional Lindblad equations. These equations rule the time evolution of density matrices of open quantum systems. The numerical techniques we present aim

at adaptively constructing a low-rank approximation of the density matrices, deriving an evolution equation for this reduced model, and using it as a surrogate model for the original evolution. Alternately, using that reduced model, we also consider and improve advanced Monte-Carlo type techniques that simulate the stochastic system of equations equivalent to the Lindblad equation. The practically relevant setting where we test our approaches arises in stabilization/control problems for quantum optics and circuits. We believe that, interestingly, our approaches may be readily adapted to problems involving the simulation and control of the evolution of density matrices in other contexts.

Bio. Claude Le Bris is a civil engineer in chief, HdR from University Paris Dauphine. His applied mathematics works have primarily been devoted to the design and analysis of numerical approaches for Physics and Mechanics. He holds a research position at Ecole des Ponts et Chaussées. He is the scientific leader of the project-team MATHERIALS at INRIA, the activity of which is focused on multiscale numerical simulation.

### Coded Caching in Wireless Networks

**Meixia Tao**

The global mobile data traffic has been shifting from voice and messages to rich content distributions, such as video streaming and application downloads. These contents are typically produced ahead of transmission and can be requested by multiple users though at possibly different times. By prefetching popular contents during off-peak times at the edge of wireless networks, such as small base stations, helper nodes, and user devices, wireless caching can alleviate peak-time network congestion and reduce user access latency. A fundamental question is what and how much gain can be leveraged by caching. In this talk, we shall investigate the gain of caching in two types of wireless networks. One is a general wireless interference network with arbitrary number of transmitters and arbitrary number of receives and with caches equipped at all the nodes. An information-theoretic study in terms of the storage-latency tradeoff will be presented. The other is a large-scale small-cell network where each small base station is equipped with a cache. We apply stochastic geometry to model, analyze, and optimize coded caching with performances characterized by average fractional offloaded traffic and average ergodic rate. Our study reveals several design insights of caching in practical wireless networks.

Bio: Meixia Tao received the B.S. degree from Fudan University, Shanghai, China, in 1999, and the Ph.D. degree from Hong Kong University of Science and Technology in 2003. She is currently a Professor with the Department of Electronic Engineering, Shanghai Jiao Tong University, China. Prior to that, she was a Member of Professional Staff at Hong Kong Applied Science and Technology Research Institute during 2003-2004, and a Teaching Fellow then an Assistant Professor at the Department of Electrical and Computer Engineering, National University of Singapore from 2004 to 2007. Her current research interests include content-centric wireless networks, wireless caching and multicasting, resource allocation, and interference coordination.

Dr. Tao is currently serving as a member of the Executive Editorial Committee of the IEEE Transactions on Wireless Communications and an Editor for the IEEE Transactions on Communications. Dr. Tao is the recipient of the IEEE Heinrich Hertz Award for Best Communications Letters in 2013 and the IEEE ComSoc Asia-Pacific Outstanding Young Researcher Award in 2009. She also receives the best paper awards from IEEE/CIC ICCC 2015 and IEEE WCSP 2012.

### Séminaire d’Automatique du plateau de Saclay : Chance-constrained optimization with tight confidence bounds.

**Mark Cannon (Department of Engineering Science, University of Oxford)**

Optimization problems involving chance constraints, i.e. constraints on decision variables that are required to hold with given probability, occur in numerous engineering contexts, but have in particular contributed to developments in robust control design over the past decade. This talk will provide a new perspective on randomized methods for solving chance-constrained programming problems based on samples of uncertain parameters, and presents a theoretical framework for sampled convex programming that encompasses analyses of constraint discarding approaches and sequential sampling approaches. We show that tight bounds on the confidence of the solution of a sampled problem meeting chance constraints apply if a randomized sample discarding strategy is employed. This suggests a solution methodology which is both computationally convenient, being based on the solutions of convex sampled optimization problems, and efficient, requiring the solution of smaller numbers of problems than existing constraint discarding and sequential methods. We illustrate the method with examples from stochastic model predictive control design.

Bio. Mark Cannon obtained the degrees of MEng in Engineering Science in 1993 and D.Phil. in Control Engineering in 1998, both from Oxford University, and SM in Mechanical Engineering in 1995 from Massachusetts Institute of Technology. He is currently Associate Professor of Engineering Science, Oxford University, and an Official Fellow of St John's College. His research interests are in robust constrained control and stochastic model predictive control, specifically: issues relating to optimization and controller design, closed loop stability and constraint satisfaction, and applications to power management in electric vehicles.

### Séminaire d’Automatique du plateau de Saclay : On generalized homogeneity and time-constrained stabilization of evolution systems.

**Andrey polyakov (NON-A team, Inria-Lille-Nord Europe)**

Homogeneity is a symmetry of an object (e.g. function or operator) with respect to some transformations (dilations). Nonlinear homogeneous ODEs (ordinary differential equations) form an important class of models of control systems. They appear as local approximations of nonlinear plant and include models of process control, nonholonomic mechanical systems, models with frictions, etc. Being non-linear the homogeneous systems demonstrate properties typical for linear systems, for example, local stability implies the global one, stable homogeneous control system is ISS (input-to-state stable) with respect to measurement noises and additive exogenous disturbance, etc.

This talk is devoted to extension of ideas of homogeneity to evolution systems in Banach/Hilbert spaces. A lot of well-known partial differential equations are homogeneous in a generalized sense, (e.g. heat, wave, Navier-Stocks, Saint-Venant, Korteweg-de Vries, fast diffusion equations). They inherit many important properties of homogeneous ODEs such as scalability of trajectories or finite-time stability in the case of negative homogeneity degree. Homogeneity allows us to design a universal control for finite-time stabilization of evolution system.

Bio. Andrey Polyakov received PhD degree from Voronezh State University (Russia) in 2005.

He was lecturer (2004-2007) and associate professor (2008-2010) of this university.

In 2007 and 2008 he was a post-doctoral research associate with Automatic Control Department of CINVESTAV (Mexico). From 2010 till 2013 he was researcher with Institute of Control Sciences of Russian Academy of Sciences. Now Andrey Polyakovis Inria researcher with NON-A team of Inria Lille Nord Europe (France). His research interests include

different problems of robust nonlinear control and estimation. He is co-author of the book "Attractive Ellipsoids in Robust Control". He is editor of International Journal of Robust and Nonlinear Control, Journal of Optimization Theory and Applications (JOTA), Automation and Remote Control.

### Séminaire d'Automatique du plateau de Saclay : Controllability of a bent 3-link magnetic microswimmer

**Pierre Lissy (CEREMADE, Universite Paris-Dauphine)**

A recent promising technique for moving a robotic micro-swimmer (in view notably of medical applications) is to apply an external magnetic field. In this talk, I will focus on a 3-link magnetic microswimmer, which consists of three rigid magnetized segments connected by two torsional springs, one of the springs being twisted, so that the swimmer is not aligned at rest. By acting on it with

an external magnetic field, the swimmer twists and moves through the surrounding fluid. After explaining some specific difficulties coming from the Low Reynolds number regime, I will explain how to model the problem thanks to a system of non-linear ODEs. By considering the external magnetic field as a control function, I will state a local partial controllability result around the equilibrium states. Then, I will propose a constructive method to find a magnetic field that allows the swimmer to move along a prescribed trajectory (tracking) in view of obtaining global partial controllability results. Finally, I will show some numerical simulations that illustrate the practical difficulties of the tracking method due to the straight positions of the swimmer. This is a joint work with Laetitia Giraldi (INRIA Sophia), Jean-Baptiste Pomet (INRIA Sophia) and Clément Moreau (ENS Cachan).

Bio: Pierre Lissy received his Ph.D degree in Applied Mathematics in 2013 under the direction of Professor Jean-Michel Coron at Université Pierre et Marie Curie. He is now Assistant Professor in Applied Mathematics at University Paris-Dauphine since 2014. His research interests include the controllability properties of partial differential equations (PDEs), with a focus on the controllability of (linear and non-linear) coupled systems of PDEs and the estimations of the cost of the control near the minimal time of control and in singular limits. He began recently to work on the controllability properties of magnetized micro-swimmers governed by non-linear ODEs.

### Séminaire d'Automatique du plateau de Saclay : Time-extremal trajectories of generic control-affine systems have at most finite-order Fuller singularities

**Francesco Boarotto (CMAP, Ecole Polytechnique)**

In this talk we show that, in fact, for the generic choice of the pair $(f_0,f_1),$ there exists an integer $N>0$ such that the control $u$ associated to any time-extremal trajectory admits at most Fuller times of order $N$. In particular, $u$ is smooth out of a set of measure zero. This is a joint work with M. Sigalotti (CMAP, Ecole Polytechnique).

Bio: Francesco Boarotto was born in Verona, Italie, in 1988. He received the Master's degree in mathematics from the University of Padou, Italie, in 2012 and the Ph.D degree from SISSA, Trieste, Italie, in 2016. Since then he has been post-doc in CMAP - Ecole Polytechnique. His research interests include geometric control theory and sub-Riemannian geometry.

### Séminaire d'Automatique du plateau de Saclay : Computation of Curvature Penalized Shortest Paths via the Fast Marching Algorithm

**Jean-Marie Mirebeau (Laboratoire de mathématique d'Orsay, Université Paris-Sud)**

Motivated by applications to motion planning and image segmentation, we consider shortest paths models with a curvature penalization, such as the Euler/Mumford elasticas, or the Reed-Shepp car with or without reverse gear. Our numerical strategy, for computing the path of minimal energy joining two given points, involves approximating these singular models using strongly anisotropic Riemannian or Finslerian metrics on the product space R^d x S^{d-1}. The associated eikonal equations are then solved via specialized variants of the Fast-Marching algorithm.

Bio. Jean-Marie Mirebeau est chargé de recherches au Laboratoire de mathématiques d'Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay. Ses travaux portent sur la résolution numérique des équations aux dérivées partielles, et en particulier sur les difficultés liées aux fortes anisotropies. C'est à dire à l'existence de directions privilégiées par le modèle, non alignées avec les axes de coordonnées. La conception de schémas pour ces modèles requiert des outils mathématiques peu communs en analyse, souvent issus de l'arithmétique et de la géométrie discrète. Son activité englobe l'étude théorique de la convergence et de la complexité des schémas numériques, leur implémentation et leur distribution en license libre, et le suivi de leur application via des collaborations académiques et industrielles. Jean-Marie Mirebeau a reçu le prix Popov 2016 pour ses contributions en théorie de l'approximation. Il était antérieurement affecté au laboratoire Ceremade de l'Université Paris-Dauphine, et a effectué sa thèse sous la direction d'Albert Cohen à l'Université Pierre et Marie Curie.

### Séminaire d'Automatique du plateau de Saclay : A relaxation result for state constrained delay differential inclusion

**Ihab Haidar (Laboratoire Quartz, ENSEA-CERGY)**

This talk is interested by delay differential inclusions in finite dimensional real space. The celebrated Filippov’s theorem is extended to this case. Then, this theorem is generalized to the case when the state variable is constrained to the closure of an open state subset. Under a new “inward pointing condition”, a relaxation result stating that the set of trajectories lying in the interior of the state constraint is dense in the set of constrained trajectories of the convexified inclusion is shown.

Bio. Ihab Haidar was born in Beirut, Lebanon, in 1983. He received the Master’s degree in mathematics from the University of Aix-Marseille 1, France, in 2008 and the Ph.D degree from the University of Montpellier 2, France, in 2011. Since then he has been post-doc in different places (Laboratoire des Signaux et Systèmes, Institut de Mathématiques de Jussieu, Laboratoire QUARTZ-ENSEA). His research interests include control theory, time delay systems and systems biology.

### Séminaire d’Automatique du plateau de Saclay : Cooperative Control of Multi-Agents: On a Sphere Manifold and in the Euclidean Space

**Wei Li (Department of Control and Systems Engineering, Nanjing University)**

The talk will discuss cooperative control of multi-agents on a sphere and in the Euclidean space. We will first consider the control law design of agents on a sphere, and analyze the stability, scaling, and geometry properties, and discuss future directions. Then, for agents evolving in the Euclidean space, we will consider coupled agents with second-order dynamics. The state of a single agent includes both position and velocity, thus generally, the agents have both velocity coupling and position couplings (VCPC); and if we consider different VCPC, then interesting yet difficult problems arise. We then discuss two aspects of analysis on consensus convergence , and future directions.

Bio. Wei Li received the Ph.D. degree in Automatic Control from Shanghai Jiao Tong University, Shanghai, China, in 2008.

From 2009 to 2010, he was a Post-Doctoral Research Associate with the Department of Electrical Engineering, The University of Texas at Dallas, Dallas, TX, USA. Since 2010, he has been an Associate Professor with the Department of Control and Systems Engineering, Nanjing University, Nanjing, China. His current research interests include robotics, autonomous mobile robots, decentralized control, cooperative control of mobile robotic agents, and wireless sensor networks. Dr. Li is an Associate Editor of Asian Journal of Control. He is a Senior Member of IEEE.