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⟩

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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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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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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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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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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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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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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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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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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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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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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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