2024

Journal articles

- titre
- Conditions for uniform ensemble output controllability, and obstruction to uniform ensemble controllability
- auteur
- Baparou Danhane, Jérôme Lohéac, Marc Jungers
- article
*Mathematical Control and Related Fields*, 2024, 14 (3), pp.1128-1175. ⟨10.3934/mcrf.2023036⟩- resume
- We consider the linear time-invariant system ẋ(t, θ) = A(θ)x(t, θ) + B(θ)u(t) with output y(t, θ) = C(θ)x(t, θ) where A, B and C are continuous matrices with respect to the constant parameter θ, which belongs to some compact set P. Given any continuous initial state datum θ → x₀(θ) and any continuous output function θ → y₁(θ), we investigate the existence of a θ-independent open loop control u such that x₀ is steered, in nite time, arbitrarily closed to y₁ with the uniform norm. When C(θ) is the identity matrix, this notion is usually termed to be uniform ensemble controllability. In this paper, we extend some result on uniform ensemble controllability to the case where C(θ) is not the identity matrix. We will also give some obstructions to ensemble controllability when the parameter set admits an interior point.
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- titre
- Ensemble controllability of parabolic type equations
- auteur
- Baparou Danhane, Jérôme Lohéac
- article
*Systems and Control Letters*, 2024, 183, pp.105683. ⟨10.1016/j.sysconle.2023.105683⟩- resume
- In this paper, we extend the concept of ensemble controllability introduced in 2009 by Jr-Shin Li and Navin Khaneja to a class of linear partial differential equation. More precisely, we consider some abstract parabolic equation, where the system depends on some unknown parameter which is assumed to belong to a compact interval. We investigate the possibility of approximatively reaching (in L²-norm) any target state from any initial state, with an open loop control. Here the initial and target states might depend on the unknown parameter, but the control is assumed to be parameter independent.
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Preprints, Working Papers, ...

- titre
- Discrete-time conewise linear systems with finitely many switches
- auteur
- Jamal Daafouz, Jérôme Lohéac, Irinel-Constantin Morarescu, Romain Postoyan
- article
- 2024
- resume
- We investigate discrete-time conewise linear systems (CLS) for which all the solutions exhibit a finite number of switches. By switches, we mean transitions of a solution from one cone to another. Our interest in this class of CLS comes from the optimization-based control of an insulin infusion model for which the fact that solutions switch finitely many times appears to be key to establish the global exponential stability of the origin. The stability analysis of this class of CLS greatly simplifies compared to general CLS as all solutions eventually exhibit linear dynamics. The main challenge is to characterize CLS satisfying this finite number of switches property. We first present general conditions in terms of set intersections for this purpose. To ease the testing of these conditions, we translate them as a non-negativity test of linear forms using Farkas lemma. As a result, the problem reduces to verify the non-negativity of a single solution to an auxiliary linear discrete-time system. Interestingly, this property differs from the classical non-negativity problem, where any solution to a system must remain non-negative (component-wise) for any non-negative initial condition, and thus requires novel tools to test it. We finally illustrate the relevance of the presented results on the optimal insulin infusion problem.
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- titre
- Sparsity and L1-optimal control for linear control systems
- auteur
- Pierre Cavaré, Marc Jungers, Jérôme Lohéac
- article
- 2024
- resume
- For linear time-invariant systems, we take an interest in sparse controls, i.e., those whose support (or L0-norm) is of minimal Lebesgue measure. We propose to extend this research to Radon measures and, in particular, to controls presenting Dirac impulses. Then, we can find a minimizer which is impulsive (and so with a null support) and prove that there is no gap when we extend the problem from L1-controls to Radon measures. To refine our research, this reasoning leads us to study the minimization problems for L1 and measure norms. We prove that impulsive controls are still pertinent for these problems: once again, we can find a minimizer and prove that there is no gap. In addition to the existence of a solution, we show that, by extending the research to Radon measures, we can find a solution of L0 and L1-minimization problems in the form of a finite linear combination of Dirac impulses, and we give an expression of the maximal number of these impulses. We also propose an algorithm in order to compute an impulsive solution of the measure norm minimization problem.
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2023

Journal articles

- titre
- A position-dependent acoustic model relevant for some active noise control applications
- auteur
- Chaouki Nacer Eddine Boultifat, Jérôme Lohéac, Mohamed Yagoubi, Philippe Chevrel
- article
*Journal of Vibration and Control*, 2023, 29 (23-24), pp.5500-5510. ⟨10.1177/10775463221136902⟩- resume
- This paper focuses on acoustic modeling for active noise control applications. The desired model must be nite dimensional and ecient numerically to meet implementation requirements. It also must be parameterized by the position considered inside the cavity. This additional characteristic is desirable for addressing specic active noise control applications such as optimizing microphone placement, estimating acoustic pressure at dierent positions of the considered cavity and nally attenuating noise level in a subarea of the cavity. The main contribution of this paper is to propose a nite-dimensional, low-order, and parameterized acoustic model of a cavity, suitable for active noise control applications. The resulting model is dened as a gray box that combines a one-dimensional analytical model of acoustic propagation, which handles the physical parameterization, and a black-box model that copes with actuator and sensor dynamics as well as modeling errors. The parameters of the proposed model are optimized in order to reproduce the frequency behavior of the real system (LS2N active noise control platform) in a frequency range. This model allows one to accurately reproduce the dynamics at any position in the considered cavity. The prediction performance of the proposed model was compared to a classical black-box model (usually used for active noise control applications) and validated experimentally using the LS2N active noise control platform. The analysis highlighted that the proposed gray-box model can predict the acoustic behavior in a great range of positions.
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- titre
- Characterizations of output controllability for LTI systems
- auteur
- Baparou Danhane, Jérôme Lohéac, Marc Jungers
- article
*Automatica*, 2023, 154, pp.111104. ⟨10.1016/j.automatica.2023.111104⟩- resume
- The objective of this paper is to make, in a simple and rigorous way, some contributions to the notion of output controllability. We first examine, for Linear Time-Invariant systems, the notion of state to output controllability, introduced in the 60s by Bertram and Sarachik. More precisely, we extend the Hautus test, well-known in the case of state controllability, to state to output controllability, and propose a controllability Gramian matrix, allowing us to build a continuous control achieving a transfer with minimal energy. We also give two other notions of output controllability, namely output to output and globally output to output controllability. For each of these two new notions, we give necessary and sufficient conditions, in terms of Kalman rank, Hautus test and Gramian matrices. All of these results are given in the framework of continuous time and discretized time systems. These results are illustrated by several numerical examples.
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- titre
- Controllability of the Stefan problem by the flatness approach
- auteur
- Blaise Colle, Jérôme Lohéac, Takéo Takahashi
- article
*Systems and Control Letters*, 2023, 174, pp.105480. ⟨10.1016/j.sysconle.2023.105480⟩- resume
- We show the boundary controllability to stationary states of the Stefan problem with two phases and in one dimension in the space variable. In the case of an initial condition that is a stationary state and for a time of control large enough, we also obtain the controllability to stationary states with the sign constraints associated with the problem. Our method is based on the flatness approach that consists in writing the solution and the controls through two outputs and their derivatives. We construct these outputs as Gevrey functions of order σ so that our solution and controls are also in a Gevrey class.
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- titre
- Controllability results for a cross diffusion system with a free boundary by a flatness approach
- auteur
- Blaise Colle, Jérôme Lohéac, Takéo Takahashi
- article
*Acta Applicandae Mathematicae*, 2023, 187, pp.14. ⟨10.1007/s10440-023-00607-0⟩- resume
- We study a one-dimensional cross diffusion system with a free boundary modeling Physical Vapor Deposition. Using the flatness approach, we show several results of boundary controllability for this system in spaces of Gevrey class functions. One of the main difficulties consists in the physical constraints on the state and on the control. More precisely, the state corresponds to volume fractions of the n + 1 chemical species and to the thickness of the film produced in the process, whereas the controls are the fluxes of the chemical species. We obtain the local controllability in the case where we apply n + 1 nonnegative controls and a controllability result for large time in the case where we apply n controls without any sign constraints. We also show in this last case that the controllability may fail for small times. We illustrate these results with some numerical simulations.
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- titre
- Flatness approach for the boundary controllability of a system of heat equations
- auteur
- Blaise Colle, Jérôme Lohéac, Takéo Takahashi
- article
*SIAM Journal on Control and Optimization*, 2023, 62 (3), pp.1766-1782. ⟨10.1137/23M1577833⟩- resume
- We study the boundary controllability of $2 × 2$ system of heat equations by using a flatness approach. According to the relation between the diffusion coefficients of the heat equation, it is known that the system can be not null-controllable or null-controllable for any $T > T_0$ where $T_0 \in [0, \infty]$. Here we recover this result in the case that $T_0 \in [0, \infty)$ by using the flatness method, and we obtain an explicit formula for the control and for the corresponding solutions. In particular, the state and the control have Gevrey regularity in time and in space.
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2022

Journal articles

- titre
- Exact noise cancellation for 1D-acoustic propagation systems
- auteur
- Jérôme Lohéac, Chaouki Nacer Eddine Boultifat, Philippe Chevrel, Mohamed Yagoubi
- article
*Mathematical Control and Related Fields*, 2022, 12 (1), pp.1-16. ⟨10.3934/mcrf.2020055⟩- resume
- This paper deals with active noise control applied to a one-dimensional acoustic propagation system. The aim here is to keep over time a zero noise level at a given point. We aim to design this control using noise measurement at some point in the spatial domain. Based on symmetry property, we are able to design a feedback boundary control allowing this fact. Moreover, using D'Alembert formula, an explicit formula of the control can be computed. Even if the focus is made on the wave equation, this approach is easily extendable to more general operators.
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- titre
- Time optimal control for a mobile robot with a communication objective
- auteur
- Jérôme Lohéac, Vineeth S. Varma, Irinel-Constantin Morarescu
- article
*Mathematics and Computers in Simulation*, 2022, 201, pp.96-120. ⟨10.1016/j.matcom.2022.05.013⟩- resume
- In this paper, we design control strategies that minimize the time required by a mobile robot to accomplish a certain task (reach a target) while transmitting/receiving a message. To better illustrate the solution we consider a simple model for the robot dynamics. The message delivery is done over a wireless network, and we account for path-loss, i.e., the transmission rate depends on the distance to the wireless antenna. In this work, we consider only one wireless antenna and disregard any shadowing phenomena. To render the problem interesting from a practical point of view we assume that the robot cannot move with innite velocity. The general problem involves a switching control signal due to the complementarity of the objectives (message transmission can require to approach the antenna situated in the opposite direction of the nal target to reach). Our minimal-time control design is based on the use of Pontryagin maximum principle. A numerical example illustrates the theoretical results.
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Book sections

- titre
- Control of parameter dependent systems
- auteur
- Martin Lazar, Jérôme Lohéac
- article
*Handbook of Numerical Analysis*, 23, pp.265-306, 2022, Numerical Control: Part A, 1570-8659. ⟨10.1016/bs.hna.2021.12.008⟩- resume
- In this chapter we provide an overview of recent progress on the problem of controllability of parameter dependent systems. We explore a different control notions successfully developed through the last decade. The aim of the control function is to steer the system to a state satisfying some properties prescribed either at some time instant T > 0 or during a given time interval. These properties may be separated with respect to parameter values and can refer just to a single system itself (e.g. greedy control), or may consider solutions corresponding to the whole parameter range (e.g. ensemble control, averaged control). In the latter case control functions are designed as parameter invariant, implying a same control is to be applied to the system independently of a particular realisation of the parameter, while in the first case controls vary along with the parameter. Beside the positive theoretical results, for each notion we provide a precise computational algorithm accompanied by a numerical example.
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Software

- titre
- FlatStefan
- auteur
- Blaise Colle, Jérôme Lohéac, Takéo Takahashi
- article
- 2022, ⟨swh:1:dir:154ea1599cad1d5d02db2d8fc877cbb331157089;origin=https://hal.archives-ouvertes.fr/hal-03889209;visit=swh:1:snp:1ce4f8288fecc8288f506ece21948bc248978b33;anchor=swh:1:rel:f837eb0dff34abf2780808f0fdd9d5a7184fe058;path=/⟩
- resume
- This provides codes related to the paper "Controllability of the Stefan problem by the flatness approach" by Blaise Colle, Jérôme Lohéac and Takéo Takahashi (hal-03721544).
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2021

Journal articles

- titre
- Contributions to output controllability for Linear Time Varying systems
- auteur
- Baparou Danhane, Jérôme Lohéac, Marc Jungers
- article
*IEEE Control Systems Letters*, 2021, 6, pp.1064-1069. ⟨10.1109/LCSYS.2021.3089642⟩- resume
- The purpose of this paper is to provide some contributions to one notion of Output controllability for Linear Time Varying (LTV) systems which is the Complete State to Output Controllability (CSOC), notion introduced in the 60s by P. E. Sarachik and G. M. Kranc. More precisely, we consider LTV systems with direct transmission of the input to the output and establish criteria to ensure the CSOC in finite time of these systems. We also give, under the assumption of CSOC in finite time, an explicit expression of a continuous control built by means of a Gramian matrix.
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- titre
- Nonnegative boundary control of 1D linear heat equations
- auteur
- Jérôme Lohéac
- article
*Vietnam Journal of Mathematics*, 2021, 49 (3), pp.845-870. ⟨10.1007/s10013-021-00497-5⟩- resume
- We consider the controllability of a one dimensional heat equation with nonnegative boundary controls. Despite the controllability in any positive time of this system, the unilateral nonnegativity control constraint causes a positive minimal controllability time. In this article, it is proved that at the minimal time, there exists a nonnegative control in the space of Radon measures, which consists of a countable sum of Dirac impulses.
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- titre
- Nonnegative control of finite-dimensional linear systems
- auteur
- Jérôme Lohéac, Emmanuel Trélat, Enrique Zuazua
- article
*Annales de l'Institut Henri Poincaré C, Analyse non linéaire*, 2021, 38 (2), pp.301-346. ⟨10.1016/j.anihpc.2020.07.004⟩- resume
- We consider the controllability problem for finite-dimensional linear autonomous control systems with nonnegative controls. Despite the Kalman condition, the unilateral nonnegativity control constraint may cause a positive minimal controllability time. When this happens, we prove that, if the matrix of the system has a real eigenvalue, then there is a minimal time control in the space of Radon measures, which consists of a finite sum of Dirac impulses. When all eigenvalues are real, this control is unique and the number of impulses is less than half the dimension of the space. We also focus on the control system corresponding to a finite-difference spatial discretization of the one-dimensional heat equation with Dirichlet boundary controls, and we provide numerical simulations.
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2020

Journal articles

- titre
- Controllability of low Reynolds numbers swimmers of ciliate type
- auteur
- Jérôme Lohéac, Takéo Takahashi
- article
*ESAIM: Control, Optimisation and Calculus of Variations*, 2020, 26, pp.31. ⟨10.1051/cocv/2019010⟩- resume
- We study the locomotion of a ciliated microorganism in a viscous incompressible fluid. We use the Blake ciliated model: the swimmer is a rigid body with tangential displacements at its boundary that allow it to propel in a Stokes fluid. This can be seen as a control problem: using periodical displacements, is it possible to reach a given position and a given orientation? We are interested in the minimal dimension d of the space of controls that allows the microorganism to swim. Our main result states the exact controllability with d = 3 generically with respect to the shape of the swimmer and with respect to the vector fields generating the tangential displacements. The proof is based on analyticity results and on the study of the particular case of spheroidal swimmer.
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- titre
- Learning control for transmission and navigation with a mobile robot under unknown communication rates
- auteur
- Lucian Buşoniu, Vineeth Varma, Jérôme Lohéac, Alexandru Codrean, Octavian Ştefan, Irinel-Constantin Morarescu, Samson Lasaulce
- article
*Control Engineering Practice*, 2020, 100, pp.104460. ⟨10.1016/j.conengprac.2020.104460⟩- resume
- In tasks such as surveying or monitoring remote regions, an autonomous robot must move while transmitting data over a wireless network with unknown, position-dependent transmission rates. For such a robot, this paper considers the problem of transmitting a data buffer in minimum time, while possibly also navigating towards a goal position. Two approaches are proposed, each consisting of a machine-learning component that estimates the rate function from samples; and of an optimal-control component that moves the robot given the current rate function estimate. Simple obstacle avoidance is performed for the case without a goal position. In extensive simulations, these methods achieve competitive performance compared to known-rate and unknown-rate baselines. A real indoor experiment is provided in which a Parrot AR.Drone 2 successfully learns to transmit the buffer.
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- titre
- Output controllability in a long-time horizon
- auteur
- Martin Lazar, Jérôme Lohéac
- article
*Automatica*, 2020, 113, pp.108762. ⟨10.1016/j.automatica.2019.108762⟩- resume
- In this article we consider a linear finite dimensional system. Our aim is to design a control such that the output of the system reach a given target at a final time T > 0. This notion is known as output controllability. We extend this notion to the one of long-time output controllability. More precisely, we consider the question: is it possible to steer the output of the system to some prescribed value in time T > 0 and then keep the output of the system at this prescribed value for all times t > T ? We provide a necessary and sufficient condition for this property to hold. Once the condition is satisfied, one can apply a feedback control that keeps the average fixed during a given time period. We also address the L2 -norm optimality of such controls. We apply our results to (long-time) averaged control problems.
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2019

Conference papers

- titre
- Dissipativeness and dissipativation of discrete-time switched linear systems
- auteur
- Marc Jungers, Francesco Ferrante, Jérôme Lohéac
- article
*CDC 2019 - 58th IEEE Conference on Decision and Control*, Dec 2019, Nice, France. pp.5760-5765- resume
- Dissipativeness of dynamical systems is a crucial notion in control theory that consolidates the link with physics. It extends Lyapunov theory for autonomous systems to open ones and formalizes the relation between frequency domain conditions and matrix inequalities in state space representation. As emphasized in the limited and recent literature on this topic, dissipativeness of hybrid or continuous-time switched systems is a not intuitive and delicate notion. This paper copes with the dissipativeness analysis of discrete-time switched linear systems. Conditions in the form of linear matrix inequalities are provided to ensure dissipativeness of such systems with arbitrary switching law. The approach relies on modal storage functions. A second contribution is to design feedback switching laws, based on a min-switching strategy related to the modal storage functions, which ensures a dissipative behaviour of the closed-loop system. Implication in terms of passivity and stability of one single switched system, paving the way to the framework of interconnected switched sub-systems are discussed, before numerical illustrations.
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- titre
- Exact noise estimation and cancellation for 1D-acoustic propagation system
- auteur
- Jérôme Lohéac, Chaouki Nacer Eddine Boultifat, Philippe Chevrel, Mohamed Yagoubi
- article
*GT Outils pour l’analyse et la synthèse de systèmes de dimension infinie, OSYDI du GdR MACS*, Centrale Nantes, Oct 2019, Nantes, France- resume
- This talk deals with active noise control applied to a one-dimensional acoustic propagation system. The aim here is to keep over time a zero noise level at a given point. Based on symmetry property, we are able to design a feedback boundary control allowing this fact. Moreover, this control only depends on past observations made at an other point in the spatial domain. Finally, using D'Alembert formula, an explicit formula of the control can be computed. Even if the focus is made on the wave equation, this approach can be easily extended to more general operators.
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2018

Journal articles

- titre
- Minimal controllability time for finite-dimensional control systems under state constraints
- auteur
- Jérôme Lohéac, Emmanuel Trélat, Enrique Zuazua
- article
*Automatica*, 2018, 96, pp.380-392. ⟨10.1016/j.automatica.2018.07.010⟩- resume
- We consider the controllability problem for finite-dimensional linear autonomous control systems, under state constraints but without imposing any control constraint. It is well known that, under the classical Kalman condition, in the absence of constraints on the state and the control, one can drive the system from any initial state to any final one in an arbitrarily small time. Furthermore, it is also well known that there is a positive minimal time in the presence of compact control constraints. We prove that, surprisingly, a positive minimal time may be required as well under state constraints, even if one does not impose any restriction on the control. This may even occur when the state constraints are unilateral, like the nonnegativity of some components of the state, for instance. Using the Brunovsky normal forms of controllable systems, we analyze this phenomenon in detail, that we illustrate by several examples. We discuss some extensions to nonlinear control systems and formulate some challenging open problems.
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Conference papers

- titre
- A parametrized reduced order model of 1D acoustic propagation system for robust spatial multi-point active noise attenuation
- auteur
- Chaouki Boultifat, Philippe Chevrel, Jérôme Lohéac, Mohamed Yagoubi
- article
*57th IEEE Conference on Decision and Control, CDC 2018*, Dec 2018, Miami Beach, FL, United States. ⟨10.1109/CDC.2018.8619060⟩- resume
- This paper discusses active noise feedback control over a predefined frequency range for a cylindrical and closed three-dimensional cavity. An analytical 1D acoustic model is obtained by averaging the pressure on surfaces that are perpendicular to the sound propagation direction. Furthermore, a parametrized rational reduced order transfer function is proposed relying on the Cauchy method of residues. Based on this new model a Multi-objective Hinf control, aiming to attenuate the effect of an unknown noise in one or several points, is designed. Achieved performances are discussed in light of different design configurations using either a single or several measurement points.
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2017

Journal articles

- titre
- A Structure Preserving Scheme for the Kolmogorov-Fokker-Planck Equation
- auteur
- Erich L Foster, Jérôme Lohéac, Minh-Binh Tran
- article
*Journal of Computational Physics*, 2017, 330, pp.319 - 339. ⟨10.1016/j.jcp.2016.11.009⟩- resume
- In this paper we introduce a numerical scheme which preserves the long time behavior of solutions to the Kolmogorov equation. The method presented is based on a self-similar change of variables technique to transform the Kolmogorov equation into a new form, such that the problem of designing structure preserving schemes, for the original equation, amounts to building a standard scheme for the transformed equation. We also present an analysis for the operator splitting technique for the self-similar method and numerical results for the described scheme.
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- titre
- Averaged controllability of parameter dependent conservative semigroups
- auteur
- Jérôme Lohéac, Enrique Zuazua
- article
*Journal of Differential Equations*, 2017, 262 (3), pp.1540 - 1574. ⟨10.1016/j.jde.2016.10.017⟩- resume
- We consider the problem of averaged controllability for parameter depending (either in a discrete or continuous fashion) control systems, the aim being to find a control, independent of the unknown parameters, so that the average of the states is controlled. We do it in the context of conservative models, both in an abstract setting and also analysing the specific example of the wave equation. Our first result is of perturbative nature. Assuming the averaging probability measure to be a small parameter-dependent perturbation (in a sense that we make precise) of an atomic measure given by a Dirac mass corresponding to a specific realisation of the system, we show that the averaged controllability property is achieved whenever the system corresponding to the support of the Dirac is controllable. Similar tools can be employed to obtain averaged versions of the so-called Ingham inequalities. Particular attention is devoted to 1d wave and Schrödinger equations in which the time-periodicity of solutions can be exploited to obtain more precise results, provided the parameters involved satisfy Diophantine conditions ensuring the lack of resonances.
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- titre
- Minimal controllability time for the heat equation under unilateral state or control constraints
- auteur
- Jérôme Lohéac, Emmanuel Trélat, Enrique Zuazua
- article
*Mathematical Models and Methods in Applied Sciences*, 2017, 27 (9), pp.1587--1644. ⟨10.1142/S0218202517500270⟩- resume
- The heat equation with homogeneous Dirichlet boundary conditions is well known to preserve nonnegativity. Besides, due to infinite velocity propagation, the heat equation is null-controllable within arbitrary small time, with controls supported in any arbitrarily open subset of the domain (or its boundary) where heat diffuses. The following question then arises naturally: can the heat dynamics be controlled from a positive initial steady-state to a positive final one, requiring that the state remains nonnegative along the controlled time-dependent trajectory? We show that this state-constrained controllability property can be achieved if the control time is large enough, but that it fails to be true in general if the control time is too short, thus showing the existence of a positive minimal controllability time. In other words, in spite of infinite velocity propagation, realizing controllability under the unilateral nonnegativity state constraint requires a positive minimal time. We establish similar results for unilateral control constraints. We give some explicit bounds on the minimal controllability time, first in 1D by using the sinusoidal spectral expansion of solutions, and then in multi-D on any bounded domain. We illustrate our results with numerical simulations, and we discuss similar issues for other control problems with various boundary conditions.
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Conference papers

- titre
- FxLMS versus H∞ Control for Broadband Acoustic Noise Attenuation in a Cavity
- auteur
- Chaouki Boultifat, Paul Loiseau, Philippe Chevrel, Jerome Loheac, Mohamed Yagoubi
- article
*IFAC 2017 : 20th World Congress of the International Federation of Automatic Control*, Jul 2017, Toulouse, France. ⟨10.1016/j.ifacol.2017.08.1277⟩- Accès au bibtex

- titre
- One-Dimensional Acoustic Propagation Model and Spatial Multi-Point Active Noise Control
- auteur
- Chaouki Boultifat, Philippe Chevrel, Jérôme Lohéac, Mohamed Yagoubi, Paul Loiseau
- article
*CDC 2017 : IEEE 56th Conference on Decision and Control*, IEEE, Dec 2017, Melbourne, Australia. ⟨10.1109/CDC.2017.8264088⟩- resume
- This paper deals with active noise control (ANC) on a broadband frequency range for a 3D-cylindrical cavity closed on two sides. A 1D acoustic model is first derived by averaging the pressure on surfaces that are perpendicular to the acoustic propagation direction. Based on the novel analytic model proposed, robust Multi-Objective Hinfty control (MOHinfty) is considered to explore the possibilities of noise attenuation not only at the error signal got from the microphone, but around. The SISO feedback control scheme considered (one microphone measuring the residual noise and one loudspeaker) has no direct information on the disturbance noise. Within this framework, the achievable attenuations both with regard to frequency and to space are discussed.
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Book sections

- titre
- Time optimal control for a perturbed Brockett integrator
- auteur
- Jérôme Lohéac, Jean-Francois Scheid
- article
*Variational Methods In Imaging and Geometric Control*, 18, , pp.454--472, 2017, Radon Series on Computational and Applied Mathematics, 978-3-11-043923-6. ⟨10.1515/9783110430394-015⟩- resume
- The aim of this work is to compute time optimal controls for a perturbation of a Brockett integrator with state constraints. Brockett integrator and its perturbations appears in many applications fields. One of them described in details in this note is the swimming of microorganisms. We present some key results for a fast and robust numerical method to compute time optimal controls for the perturbation of a Brockett integrator. This numerical method is based on explicit formulae of time optimal controls for the Brockett integrator. The methodology presented in this work is applied to the time optimal control of a micro-swimmer.
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2016

Journal articles

- titre
- From averaged to simultaneous controllability
- auteur
- Jérôme Lohéac, Enrique Zuazua
- article
*Annales de la Faculté des Sciences de Toulouse. Mathématiques.*, 2016, 25 (4), pp.785 - 828. ⟨10.5802/afst.1511⟩- resume
- We consider a linear finite dimensional control system depending on unknown parameters. We aim to design controls, independent of the parameters, to control the system in some optimal sense. We discuss the notions of averaged control, according to which one aims to control only the average of the states with respect to the unknown parameters, and the notion of simultaneous control in which the goal is to control the system for all values of these parameters. We show how these notions are connected through a penalization process. Roughly, averaged control is a relaxed version of the simultaneous control property, in which the differences of the states with respect to the various parameters are left free, while simultaneous control can be achieved by reinforcing the averaged control property by penalizing these differences. We show however that these two notions require of different rank conditions on the matrices determining the dynamics and the control. When the stronger conditions for simultaneous control are fulfilled, one can obtain the later as a limit, through this penalization process, out of the averaged control property.
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- titre
- Norm saturating property of time optimal controls for wave-type equations
- auteur
- Jérôme Lohéac, Enrique Zuazua
- article
*IFAC-PapersOnLine*, 2016, 49 (8), pp.37 - 42. ⟨10.1016/j.ifacol.2016.07.415⟩- resume
- We consider a time optimal control problem with point target for a class of infinite dimensional systems governed by abstract wave operators. In order to ensure the existence of a time optimal control, we consider controls of energy bounded by a prescribed constant E > 0. Even when this control constraint is absent, in many situations, due to the hyperbolicity of the system under consideration, a target point cannot be reached in arbitrarily small time and there exists a minimal universal controllability time T * > 0, so that for every points y 0 and y 1 and every time T > T * , there exists a control steering y 0 to y 1 in time T. Simultaneously this may be impossible if T < T * for some particular choices of y 0 and y 1. In this note we point out the impact of the strict positivity of the minimal time T * on the structure of the norm of time optimal controls. In other words, the question we address is the following: If τ is the minimal time, what is the L 2-norm of the associated time optimal control? For different values of y 0 , y 1 and E, we can have τ T * or τ > T *. If τ > T * , the time optimal control is unique, given by an adjoint problem and its L 2-norm is E, in the classical sense. In this case, the time optimal control is also a norm optimal control. But when τ < T * , we show, analyzing the string equation with Dirichlet boundary control, that, surprisingly, there exist time optimal controls which are not of maximal norm E.
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2015

Preprints, Working Papers, ...

- titre
- Time optimal swimmers and Brockett integrator
- auteur
- Jérôme Lohéac, Jean-Francois Scheid
- article
- 2015
- resume
- The aim of this work is to compute time optimal controls for micro-swimmers. The action of swimming is seen as a control problem. More precisely, given an initial position and a target position, can the swimmer move to the target by changing its shape. The motion of the swimmer in the fluid results from the fluid-structure interaction. For micro-swimmers, the fluid equations in consideration are the stationnary Stokes equations. The way of swimming can be described by the following steps: 1. the swimmer modifies its shape, 2. this modification creates a velocity field in the fluid, 3. the fluid velocity acts on the swimmer as a force, 4. the fluid force moves the swimmer. Of course things are not so distinct and the swimming is a highly coupled nonlinear control problem. In this note, we present some key results for a fast numerical method to compute time optimal controls for axi-symmetric micro-swimmers. This numerical method is based on explicit formulae of time optimal controls for the Brockett integrator which is a system approaching the dynamic of the swimmer.
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2014

Journal articles

- titre
- Controllability of 3D low Reynolds swimmers
- auteur
- Jérôme Lohéac, Alexandre Munnier
- article
*ESAIM: Control, Optimisation and Calculus of Variations*, 2014, 20 (1), pp.236--268. ⟨10.1051/cocv/2013063⟩- resume
- In this article, we consider a swimmer (i.e. a self-deformable body) immersed in a fluid, the flow of which is governed by the stationary Stokes equations. This model is relevant for studying the locomotion of microorganisms or micro robots for which the inertia effects can be neglected. Our first main contribution is to prove that any such microswimmer has the ability to track, by performing a sequence of shape changes, any given trajectory in the fluid. We show that, in addition, this can be done by means of arbitrarily small body deformations that can be superimposed to any preassigned sequence of macro shape changes. Our second contribution is to prove that, when no macro deformations are prescribed, tracking is generically possible by means of shape changes obtained as a suitable combination of only four elementary deformations. Eventually, still considering finite dimensional deformations, we state results about the existence of optimal swimming strategies for a wide class of cost functionals.
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2013

Journal articles

- titre
- Controllability and time optimal control for low Reynolds numbers swimmers
- auteur
- Jérôme Lohéac, Jean-François Scheid, Marius Tucsnak
- article
*Acta Applicandae Mathematicae*, 2013, 123, pp.175--200. ⟨10.1007/s10440-012-9760-9⟩- resume
- The aim of this paper is to tackle the self-propelling at low Reynolds number by using tools coming from control theory. More precisely we first address the controllability problem: "Given two arbitrary positions, does it exist "controls" such that the body can swim from one position to another, with null initial and final deformations?". We consider a spherical object surrounded by a viscous incompressible fluid filling the remaining part of the three dimensional space. The object is undergoing radial and axi-symmetric deformations in order to propel itself in the fluid. Since we assume that the motion takes place at low Reynolds number, the fluid is governed by the Stokes equations. In this case, the governing equations can be reduced to a finite dimensional control system. By combining perturbation arguments and Lie brackets computations, we establish the controllability property. Finally we study the time optimal control problem for a simplified system. We derive the necessary optimality conditions by using the Pontryagin maximum principle. In several particular cases we are able to compute the explicit form of the time optimal control and to investigate the variation of optimal solutions with respect to the number of inputs.
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- titre
- Maximum principle and bang-bang property of time optimal controls for Schrödinger type systems
- auteur
- Jérôme Lohéac, Marius Tucsnak
- article
*SIAM Journal on Control and Optimization*, 2013, 51 (5), pp.4016--4038. ⟨10.1137/120872437⟩- resume
- We consider the time optimal control problem, with a point target, for a class of infinite dimensional systems with a dynamics governed by an abstract Schrödinger type equation. The main results establish a Pontryagyn type maximum principle and give sufficient conditions for the bang-bang property of optimal controls. The results are then applied to some systems governed by partial differential equations. The paper ends by a discussion of possible extensions and by stating some open problems.
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- titre
- Time optimal control for a nonholonomic system with state constraint
- auteur
- Jérôme Lohéac, Jean-François Scheid
- article
*Mathematical Control and Related Fields*, 2013, 3 (2), pp.185-208. ⟨10.3934/mcrf.2013.3.185⟩- resume
- The aim of this paper is to tackle the time optimal controllability of an (n+1)-dimensional nonholonomic integrator. In the optimal control problem we consider, the state variables are subject to a bound constraint. We give a full description of the optimal control and optimal trajectories are explicitly obtained. The optimal trajectories we construct, lie in a 2-dimensional plane and they are composed of arcs of circle.
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2012

Theses

- titre
- Time optimal control and low Reynolds number swimming
- auteur
- Jérôme Lohéac
- article
- Optimisation et contrôle [math.OC]. Université de Lorraine, 2012. Français. ⟨NNT : 2012LORR0336⟩
- resume
- This thesis is divided in two parts. The main tool of this work is time optimal control. We first consider the Pontryagin maximum principle for control system of finite dimension. After that, we give an application of this principle for the Brockett integrator with state constraints. Then, we study an extension of the Pontryagin maximum principle in the case of infinite dimensional systems. More precisely, this extension concerns the case of exactly controllable systems in any time. For instance, this can be the Schrödinger equation with internal control. Especially under some condition of approximate controllability, we can show the existence of a bang-bang control defined on a time set of positive measure. In the second part, we study the problem of swimming at low Reynolds number. A convenient physical model allows us to formulate it under the form of a control problem. We then get a controllability result on this problem. More precisely, we will show that whatever the shape of the swimmer is, the swimmer can slightly modify its shape in order to steer any prescribed trajectory. To complete this part, we consider the case of an axi-symmetric swimmer. The results of the first part allow us to find an optimal time control.
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2010

Preprints, Working Papers, ...

- titre
- A Lagrange-multiplier approach for the numerical simulation of an inextensible membrane or thread immersed in a fluid
- auteur
- Jocelyn Etienne, Jérôme Lohéac, Pierre Saramito
- article
- 2010
- resume
- The inextensibility constraint is encountered in many physical problems involving thin solids interacting with a fluid. It is generally imposed in numerical simulations by means of a penalty method. Here, we propose a novel saddle-point approach allowing to impose it through a Lagrange multiplier defined on the thin structure, the tension. The functional analysis of the problem allows to determine which boundary conditions are needed for this problem. The forces originating from the structure appear as a boundary condition for the fluid problem, defined on a moving boundary which represents the structure. The problem is discretised with mixed finite elements. The mesh of the thin solid is included in the mesh of the bulk, and is advected by its velocity in the course of time iterations. The appropriate choice of the finite element spaces for this mixed approach is discussed and it is shown that boundary conditions on the thin structure edges impact on this choice. Numerical tests are performed which demonstrate the convergence and robustness of the method. These examples include the simulation of a closed, membrane bound object (a vesicle) and of a filament or flag in a large Reynolds-number flow.
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2009

Journal articles

- titre
- Robust numerical schemes for anisotropic diffusion problems, a first step for turbulence modeling in Lagrangian hydrodynamics
- auteur
- Julien Dambrine, Philippe Hoch, Raphaël Kuate, Jérôme Lohéac, Jérôme Metral, Bernard Rebourcet, Lisl Weynans
- article
*ESAIM: Proceedings*, 2009, CEMRACS 2008 - Modelling and Numerical Simulation of Complex Fluids, 28, pp.80-99. ⟨10.1051/proc/2009040⟩- resume
- Numerous systems of conservation laws are discretized on Lagrangian meshes where cells nodes move with matter. For complex applications, cells shape or aspect ratio often do not insure sufficient accuracy to provide an acceptable numerical solution and use of ALE technics is necessary. Here we are interested with conduction phenomena depending on velocity derivatives coming from the resolution of gas dynamics equations. For that, we propose the study of a mock of second order turbulent mixing model combining an elliptical part and an hyperbolic kernel. The hyperbolic part is approximated by finite-volume centered scheme completed by a remapping step see [7]. A major part of this paper is the discretization of the anisotropic parabolic equation on polygonal distorted mesh. It is based on the scheme described in [9] ensuring the positivity of the numerical solution. We propose an alternative based on the partitioning of polygons in triangles. We show some preliminary results on a weak coupling of hydrodynamics and parabolic equation whose tensor diffusion coefficient depends on Reynolds stresses.
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