Publications

2020

• Quality Gain Analysis of the Weighted Recombination Evolution Strategy on General Convex Quadratic Functions
  
  • Akimoto Youhei
  • Auger Anne
  • Hansen Nikolaus
  Theoretical Computer Science, Elsevier, 2020, 832, pp.42-67. Quality gain is the expected relative improvement of the function value in a single step of a search algorithm. Quality gain analysis reveals the dependencies of the quality gain on the parameters of a search algorithm, based on which one can derive the optimal values for the parameters. In this paper, we investigate evolution strategies with weighted recombination on general convex quadratic functions. We derive a bound for the quality gain and two limit expressions of the quality gain. From the limit expressions, we derive the optimal recombination weights and the optimal step-size, and find that the optimal recombination weights are independent of the Hessian of the objective function. Moreover, the dependencies of the optimal parameters on the dimension and the population size are revealed. Differently from previous works where the population size is implicitly assumed to be smaller than the dimension, our results cover the population size proportional to or greater than the dimension. Simulation results show the optimal parameters derived in the limit approximates the optimal values in non-asymptotic scenarios. (10.1016/j.tcs.2018.05.015)
  DOI : 10.1016/j.tcs.2018.05.015
• On the Turnpike Property and the Receding-Horizon Method for Linear-Quadratic Optimal Control Problems
  
  • Breiten Tobias
  • Pfeiffer Laurent
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2020, 58 (2), pp.26. Optimal control problems with a very large time horizon can be tackled with the Receding Horizon Control (RHC) method, which consists in solving a sequence of optimal control problems with small prediction horizon. The main result of this article is the proof of the exponential convergence (with respect to the prediction horizon) of the control generated by the RHC method towards the exact solution of the problem. The result is established for a class of infinite-dimensional linear-quadratic optimal control problems with time-independent dynamics and integral cost. Such problems satisfy the turnpike property: the optimal trajectory remains most of the time very close to the solution to the associated static optimization problem. Specific terminal cost functions, derived from the Lagrange multiplier associated with the static optimization problem, are employed in the implementation of the RHC method. (10.1137/18M1225811)
  DOI : 10.1137/18M1225811
• Variance reduction for Markov chains with application to MCMC
  
  • Belomestny D
  • Iosipoi L
  • Moulines E
  • Naumov A
  • Samsonov S
  Statistics and Computing, Springer Verlag (Germany), 2020. In this paper we propose a novel variance reduction approach for additive functionals of Markov chains based on minimization of an estimate for the asymptotic variance of these functionals over suitable classes of control variates. A distinctive feature of the proposed approach is its ability to significantly reduce the overall finite sample variance. This feature is theoretically demonstrated by means of a deep non asymptotic analysis of a variance reduced functional as well as by a thorough simulation study. In particular we apply our method to various MCMC Bayesian estimation problems where it favourably compares to the existing variance reduction approaches.
• Multiscale population dynamics in reproductive biology: singular perturbation reduction in deterministic and stochastic models
  
  • Bonnet Celine
  • Chahour Keltoum
  • Clément Frédérique
  • Postel Marie
  • Yvinec Romain
  ESAIM: Proceedings and Surveys, EDP Sciences, 2020, 67, pp.72-99. In this study, we describe different modeling approaches for ovarian follicle population dynamics, based on either ordinary (ODE), partial (PDE) or stochastic (SDE) differential equations, and accounting for interactions between follicles. We put a special focus on representing the populationlevel feedback exerted by growing ovarian follicles onto the activation of quiescent follicles. We take advantage of the timescale difference existing between the growth and activation processes to apply model reduction techniques in the framework of singular perturbations. We first study the linear versions of the models to derive theoretical results on the convergence to the limit models. In the nonlinear cases, we provide detailed numerical evidence of convergence to the limit behavior. We reproduce the main semi-quantitative features characterizing the ovarian follicle pool, namely a bimodal distribution of the whole population, and a slope break in the decay of the quiescent pool with aging. (10.1051/proc/202067006)
  DOI : 10.1051/proc/202067006
• A game theory approach to the existence and uniqueness of nonlinear Perron-Frobenius eigenvectors
  
  • Akian Marianne
  • Gaubert Stéphane
  • Hochart Antoine
  Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2020, 40, pp.207--231. We establish a generalized Perron-Frobenius theorem, based on a combinatorial criterion which entails the existence of an eigenvector for any nonlinear order-preserving and positively homogeneous map $f$ acting on the open orthant $\mathbb{R}_{\scriptscriptstyle >0}^n$. This criterion involves dominions, i.e., sets of states that can be made invariant by one player in a two-person game that only depends on the behavior of $f$ "at infinity". In this way, we characterize the situation in which for all $\alpha, \beta > 0$, the "slice space" $\mathcal{S}_\alpha^\beta := \{ x \in \mathbb{R}_{\scriptscriptstyle >0}^n \mid \alpha x \leq f(x) \leq \beta x \}$ is bounded in Hilbert's projective metric, or, equivalently, for all uniform perturbations $g$ of $f$, all the orbits of $g$ are bounded in Hilbert's projective metric. This solves a problem raised by Gaubert and Gunawardena (Trans. AMS, 2004). We also show that the uniqueness of an eigenvector is characterized by a dominion condition, involving a different game depending now on the local behavior of $f$ near an eigenvector. We show that the dominion conditions can be verified by directed hypergraph methods. We finally illustrate these results by considering specific classes of nonlinear maps, including Shapley operators, generalized means and nonnegative tensors. (10.3934/dcds.2020009)
  DOI : 10.3934/dcds.2020009
• Accuracy assessment of the Non-Ideal Computational Fluid Dynamics model for siloxane MDM from the open-source SU2 suite
  
  • Gori Giulio
  • Zocca Marta
  • Cammi Giorgia
  • Spinelli Andrea
  • Congedo Pietro Marco
  • Guardone Alberto
  European Journal of Mechanics - B/Fluids, Elsevier, 2020, 79, pp.109-120. The first-ever accuracy assessment of a computational model for Non-Ideal Compressible-Fluid Dynamics (NICFD) flows is presented. The assessment relies on a comparison between numerical predictions, from the open-source suite SU2, and pressure and Mach number measurements of compressible fluid flows in the non-ideal regime. Namely, measurements regard supersonic flows of siloxane MDM (Octamethyltrisiloxane, C 8 H 24 O 2 Si 3) vapor expanding along isentropes in the close proximity of the liquid-vapor saturation curve. The model accuracy assessment takes advantage of an Uncertainty Quantification (UQ) analysis, to compute the variability of the numerical solution with respect the uncertainties affecting the test-rig operating conditions. This allows for an uncertainty-based assessment of the accuracy of numerical predictions. The test set is representative of typical operating conditions of Organic Rankine Cycle systems and it includes compressible flows expanding through a converging-diverging nozzle in mildly-to-highly non-ideal conditions. All the considered flows are well represented by the computational model. Therefore, the reliability of the numerical implementation and the predictiveness of the NICFD model are confirmed. (10.1016/j.euromechflu.2019.08.014)
  DOI : 10.1016/j.euromechflu.2019.08.014
• On Invariance and Linear Convergence of Evolution Strategies with Augmented Lagrangian Constraint Handling
  
  • Atamna Asma
  • Auger Anne
  • Hansen Nikolaus
  Theoretical Computer Science, Elsevier, 2020, 832, pp.68-97. In the context of numerical constrained optimization, we investigate stochastic algorithms, in particular evolution strategies, handling constraints via augmented Lagrangian approaches. In those approaches, the original constrained problem is turned into an unconstrained one and the function optimized is an augmented Lagrangian whose parameters are adapted during the optimization. The use of an augmented Lagrangian however breaks a central invariance property of evolution strategies, namely invariance to strictly increasing transformations of the objective function. We formalize nevertheless that an evolution strategy with augmented Lagrangian constraint handling should preserve invariance to strictly increasing affine transformations of the objective function and the scaling of the constraints—a subclass of strictly increasing transformations. We show that this invariance property is important for the linear convergence of these algorithms and show how both properties are connected. (10.1016/j.tcs.2018.10.006)
  DOI : 10.1016/j.tcs.2018.10.006
• A new McKean-Vlasov stochastic interpretation of the parabolic-parabolic Keller-Segel model: The one-dimensional case
  
  • Tomasevic Milica
  • Talay Denis
  Bernoulli, Bernoulli Society for Mathematical Statistics and Probability, 2020, 26 (2), pp.1323-1353. In this paper we analyze a stochastic interpretation of the one-dimensional parabolic-parabolic Keller-Segel system without cut-off. It involves an original type of McKean-Vlasov interaction kernel. At the particle level, each particle interacts with all the past of each other particle by means of a time integrated functional involving a singular kernel. At the mean-field level studied here, the McKean-Vlasov limit process interacts with all the past time marginals of its probability distribution in a similarly singular way. We prove that the parabolic-parabolic Keller-Segel system in the whole Euclidean space and the corresponding McKean-Vlasov stochastic differential equation are well-posed for any values of the parameters of the model. (10.3150/19-BEJ1158)
  DOI : 10.3150/19-BEJ1158
• Multimode communication through the turbulent atmosphere
  
  • Borcea Liliana
  • Garnier Josselin
  • Sølna Knut
  Journal of the Optical Society of America. A Optics, Image Science, and Vision, Optical Society of America, 2020, 37 (5), pp.720. A central question in free-space optical communications is how to improve the transfer of information between a transmitter and a receiver. The capacity of the communication channel can be increased by multiplexing of independent modes using either: (1) the multiple-input–multiple-output (MIMO) approach, where communication is done with modes obtained from the singular value decomposition of the transfer matrix from the transmitter array to the receiver array, or (2) the orbital angular momentum (OAM) approach, which uses vortex beams that carry angular momenta. In both cases, the number of usable modes is limited by the finite aperture of the transmitter and receiver, and the effect of the turbulent atmosphere. The goal of this paper is twofold: first, we show that the MIMO and OAM multiplexing schemes are closely related. Specifically, in the case of circular apertures, the leading singular vectors of the transfer matrix, which are useful for communication, are essentially the same as the commonly used Laguerre–Gauss vortex beams, provided these have a special radius that depends on the wavelength, the distance from the transmitter to the receiver, and the ratio of the radii of their apertures. Second, we characterize the effect of atmospheric turbulence on the communication modes using the phase screen method put in the mathematical framework of beam propagation in random media. (10.1364/JOSAA.384007)
  DOI : 10.1364/JOSAA.384007
• The boundary of random planar maps via looptrees
  
  • Kortchemski Igor
  • Richier Loïc
  Annales de la Faculté des Sciences de Toulouse. Mathématiques., Université Paul Sabatier _ Cellule Mathdoc, 2020, 29 (2), pp.391-430. (10.5802/afst.1636)
  DOI : 10.5802/afst.1636
• Parametric inference for diffusions observed at stopping times
  
  • Gobet Emmanuel
  • Stazhynski Uladzislau
  Electronic Journal of Statistics, Shaker Heights, OH : Institute of Mathematical Statistics, 2020, 14 (1). In this paper we study the problem of parametric inference for multidimensional diffusions based on observations at random stopping times. We work in the asymptotic framework of high frequency data over a fixed horizon. Previous works on the subject (such as [Doh87, GJ93, Gob01, AM04] among others) consider only deterministic, strongly predictable or random, independent of the process, observation times, and do not cover our setting. Under mild assumptions we construct a consistent sequence of estimators, for a large class of stopping time observation grids (studied in [GL14, GS18]). Further we carry out the asymptotic analysis of the estimation error and establish a Central Limit Theorem (CLT) with a mixed Gaussian limit. In addition, in the case of a 1-dimensional parameter, for any sequence of estimators verifying CLT conditions without bias, we prove a uniform a.s. lower bound on the asymptotic variance, and show that this bound is sharp. (10.1214/20-EJS1708)
  DOI : 10.1214/20-EJS1708
• Model Reduction for Large-Scale Earthquake Simulation in an Uncertain 3D Medium
  
  • Sochala Pierre
  • de Martin Florent
  • Le Maitre Olivier
  International Journal for Uncertainty Quantification, Begell House Publishers, 2020, 10 (2), pp.101-127. In this paper, we are interested in the seismic wave propagation into an uncertain medium. To this end, we performed an ensemble of 400 large-scale simulations that requires 4 million core-hours of CPU time. In addition to the large computational load of these simulations, solving the uncertainty propagation problem requires dedicated procedures to handle the complexities inherent to large data set size and the low number of samples. We focus on the peak ground motion at the free surface of the 3D domain, and our analysis utilizes a surrogate model combining two key ingredients for complexity mitigation: i) a dimension reduction technique using empirical orthogonal basis functions and ii) a functional approximation of the uncertain reduced coordinates by polynomial chaos expansions. We carefully validate the resulting surrogate model by estimating its predictive error using bootstrap, truncation, and cross-validation procedures. The surrogate model allows us to compute various statistical information of the uncertain prediction, including marginal and joint probability distributions, interval probability maps, and 2D fields of global sensitivity indices. (10.1615/Int.J.UncertaintyQuantification.2020031165)
  DOI : 10.1615/Int.J.UncertaintyQuantification.2020031165
• Multipoint formulas for scattered far field in multidimensions
  
  • Novikov Roman
  Inverse Problems, IOP Publishing, 2020, 36 (9), pp.095001(12 pp.). We give asymptotic formulas for finding the scattering amplitude at fixed frequency and angles (scattered far field) from the scattering wave function given at $n$ points in dimension $d\geq 2$. These formulas are explicit and their precision is proportional to $n$. To our knowledge these formulas are new already for $n\geq 2$. (10.1088/1361-6420/aba891)
  DOI : 10.1088/1361-6420/aba891
• A convergent hierarchy of non-linear eigenproblems to compute the joint spectral radius of nonnegative matrices
  
  • Gaubert Stéphane
  • Stott Nikolas
  Mathematical Control and Related Fields, AIMS, 2020, 10 (3), pp.573-590. We show that the joint spectral radius of a finite collection of nonnegative matrices can be bounded by the eigenvalue of a non-linear operator. This eigenvalue coincides with the ergodic constant of a risk-sensitive control problem, or of an entropy game, in which the state space consists of all switching sequences of a given length. We show that, by increasing this length, we arrive at a convergent approximation scheme to compute the joint spectral radius. The complexity of this method is exponential in the length of the switching sequences, but it is quite insensitive to the size of the matrices, allowing us to solve very large scale instances (several matrices in dimensions of order 1000 within a minute). An idea of this method is to replace a hierarchy of optimization problems, introduced by Ahmadi, Jungers, Parrilo and Roozbehani, by a hierarchy of nonlinear eigenproblems. To solve the latter eigenproblems, we introduce a projective version of Krasnoselskii-Mann iteration. This method is of independent interest as it applies more generally to the nonlinear eigenproblem for a monotone positively homogeneous map. Here, this method allows for scalability by avoiding the recourse to linear or semidefinite programming techniques. (10.3934/mcrf.2020011)
  DOI : 10.3934/mcrf.2020011
• Intensity fluctuations in random waveguides
  
  • Garnier Josselin
  Communications in Mathematical Sciences, International Press, 2020, 18 (4), pp.947-971. (10.4310/CMS.2020.v18.n4.a3)
  DOI : 10.4310/CMS.2020.v18.n4.a3
• Wave Propagation in Randomly Perturbed Weakly Coupled Waveguides
  
  • Borcea Liliana
  • Garnier Josselin
  Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Society for Industrial and Applied Mathematics, 2020, 18 (1), pp.44-78. We present an analysis of wave propagation in a two step-index, parallel waveguide system. The goal is to quantify the effect of scattering at randomly perturbed interfaces between the guiding layers of high index of refraction and the host medium. The analysis is based on the expansion of the solution of the wave equation in a complete set of guided, radiation and evanescent modes with amplitudes that are random fields, due to scattering. We obtain a detailed characterization of these amplitudes and thus quantify the transfer of power between the two waveguides in terms of their separation distance. The results show that, no matter how small the fluctuations of the interfacesare, they have significant effect at sufficiently large distance of propagation, which manifests in two ways: The first effect is well known and consists of power leakage from the guided modes to the radiation ones. The second effect consists of blurring of the periodic transfer of power between the waveguides and the eventual equipartition of power. Its quantification is the main practical result ofthe paper. (10.1137/18M1230591)
  DOI : 10.1137/18M1230591
• High-Resolution Interferometric Synthetic Aperture Imaging in Scattering Media
  
  • Borcea Liliana
  • Garnier Josselin
  SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2020, 13 (1), pp.291-316. The goal of synthetic aperture imaging is to estimate the reflectivity of a remoteregion of interest by processing data gathered with a moving sensor which emits periodically a signaland records the backscattered wave. We introduce and analyze a high-resolution interferometric method for synthetic aperture imaging through an unknown scattering medium which distorts thewave. The method builds on the coherent interferometric (CINT) approach which uses empiricalcross-correlations of the measurements to mitigate the distortion, at the expense of a loss of resolutionof the image. The new method shows that, while mitigating the wave distortion, it is possible toobtain a robust and sharp estimate of the modulus of the Fourier transform of the reflectivity function.A high-resolution image can then be obtained by a phase retrieval algorithm. (10.1137/19M1272470)
  DOI : 10.1137/19M1272470
• Support optimization in additive manufacturing for geometric and thermo-mechanical constraints
  
  • Allaire Grégoire
  • Bihr Martin
  • Bogosel Beniamin
  Structural and Multidisciplinary Optimization, Springer Verlag, 2020, 61, pp.2377-2399. Supports are often required to safely complete the building of complicated structures by additive manufacturing technologies. In particular, supports are used as scaffoldings to reinforce overhanging regions of the structure and/or are necessary to mitigate the thermal deformations and residual stresses created by the intense heat flux produced by the source term (typically a laser beam). However, including supports increase the fabrication cost and their removal is not an easy matter. Therefore, it is crucial to minimize their volume while maintaining their efficiency. Based on earlier works, we propose here some new optimization criteria. First, simple geometric criteria are considered like the projected area and the volume of supports required for overhangs: they are minimized by varying the structure orientation with respect to the baseplate. In addition, an accessibility criterion is suggested for the removal of supports, which can be used to forbid some parts of the structure to be supported. Second, shape and topology optimization of supports for compliance minimization is performed. The novelty comes from the applied surface loads which are coming either from pseudo gravity loads on overhanging parts or from equivalent thermal loads arising from the layer by layer building process. Here, only the supports are optimized, with a given non-optimizable structure, but of course many generalizations are possible, including optimizing both the structure and its supports. Our optimization algorithm relies on the level set method and shape derivatives computed by the Hadamard method. Numerical examples are given in 2-d and 3-d.
• Sparse recovery for inverse potential problems in divergence form
  
  • Baratchart Laurent
  • Villalobos Guillén Cristóbal
  • Hardin Douglas
  • Leblond Juliette
  , 2020, 1476, pp.012009. We discuss recent results on sparse recovery for inverse potential problem with source term in divergence form. The notion of sparsity which is set forth is measure-theoretic, namely pure 1-unrectifiability of the support. The theory applies when a superset of the support is known to be slender, meaning it has measure zero and all connected components of its complement has infinite measure in R^3. We also discuss open issues in the non-slender case. (10.1088/1742-6596/1476/1/012009)
  DOI : 10.1088/1742-6596/1476/1/012009
• The tropicalization of the entropic barrier
  
  • Allamigeon Xavier
  • Aznag Abdellah
  • Gaubert Stéphane
  • Hamdi Yassine
  , 2020. The entropic barrier, studied by Bubeck and Eldan (Proc. Mach. Learn. Research, 2015), is a self-concordant barrier with asymptotically optimal self-concordance parameter. In this paper, we study the tropicalization of the central path associated with the entropic barrier, i.e., the logarithmic limit of this central path for a parametric family of linear programs defined over the field of Puiseux series. Our main result is that the tropicalization of the entropic central path is a piecewise linear curve which coincides with the tropicalization of the logarithmic central path studied by Allamigeon et al. (SIAM J. Applied Alg. Geom., 2018). One consequence is that the number of linear pieces in the tropical entropic central path can be exponential in the dimension and the number of inequalities defining the linear program.
• A MOMENT CLOSURE BASED ON A PROJECTION ON THE BOUNDARY OF THE REALIZABILITY DOMAIN: 1D CASE
  
  • Pichard Teddy
  Kinetic and Related Models, AIMS, 2020, 13 (6), pp.1243-1280. This work aims to develop and test a projection technique for the construction of closing equations of moment systems. One possibility to define such a closure consists in reconstructing an underlying kinetic distribution from a vector of moments, then expressing the closure based on this reconstructed function. Exploiting the geometry of the realizability domain, i.e. the set of moments of positive distribution function, we decompose any realizable vectors into two parts, one corresponding to the moments of a chosen equilibrium function, and one obtain by a projection onto the boundary of the realizability domain in the direction of equilibrium function. A realizable closure of both of these parts are computed with standard techniques providing a realizable closure for the full system. This technique is tested for the reduction of a radiative transfer equation in slab geometry. (10.3934/xx.xx.xx.xx)
  DOI : 10.3934/xx.xx.xx.xx
• A second order analysis of McKean-Vlasov semigroups
  
  • Arnaudon Marc
  • del Moral Pierre
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2020. We propose a second order differential calculus to analyze the regularity and the stability properties of the distribution semigroup associated with McKean-Vlasov diffusions. This methodology provides second order Taylor type expansions with remainder for both the evolution semigroup as well as the stochastic flow associated with this class of nonlinear diffusions. Bismut-Elworthy-Li formulae for the gradient and the Hessian of the integro-differential operators associated with these expansions are also presented. The article also provides explicit Dyson-Phillips expansions and a refined analysis of the norm of these integro-differential operators. Under some natural and easily verifiable regularity conditions we derive a series of exponential decays inequalities with respect to the time horizon. We illustrate the impact of these results with a second order extension of the Alekseev-Gröbner lemma to nonlinear measure valued semigroups and interacting diffusion flows. This second order perturbation analysis provides direct proofs of several uniform propagation of chaos properties w.r.t. the time parameter, including bias, fluctuation error estimate as well as exponential concentration inequalities. (10.1214/20-AAP1568)
  DOI : 10.1214/20-AAP1568
• Surface waves in a channel with thin tunnels and wells at the bottom: non-reflecting underwater tomography
  
  • Chesnel Lucas
  • Nazarov Sergei
  • Taskinen Jari
  Asymptotic Analysis, IOS Press, 2020. We consider the propagation of surface water waves in a straight planar channel perturbed at the bottom by several thin curved tunnels and wells. We propose a method to construct non reflecting underwater topographies of this type at an arbitrary prescribed wave number. To proceed, we compute asymptotic expansions of the diffraction solutions with respect to the small parameter of the geometry taking into account the existence of boundary layer phenomena. We establish error estimates to validate the expansions using advances techniques of weighted spaces with detached asymptotics. In the process, we show the absence of trapped surface waves for perturbations small enough. This analysis furnishes asymptotic formulas for the scattering matrix and we use them to determine underwater topographies which are non-reflecting. Theoretical and numerical examples are given.
• A metric interpretation of the geodesic curvature in the Heisenberg group
  
  • Kohli Mathieu
  Journal of Dynamical and Control Systems, Springer Verlag, 2020, 26 (1), pp.159–174. In this paper we study the notion of geodesic curvature of smooth horizontal curves parametrized by arc lenght in the Heisenberg group, that is the simplest sub-Riemannian structure. Our goal is to give a metric interpretation of this notion of geodesic curvature as the first corrective term in the Taylor expansion of the distance between two close points of the curve. (10.1007/s10883-019-09444-7)
  DOI : 10.1007/s10883-019-09444-7
• Approximate and exact controllability of linear difference equations
  
  • Chitour Yacine
  • Mazanti Guilherme
  • Sigalotti Mario
  Journal de l'École polytechnique — Mathématiques, École polytechnique, 2020, 7, pp.93--142. In this paper, we study approximate and exact controllability of the linear difference equation $x(t) = \sum_{j=1}^N A_j x(t - \Lambda_j) + B u(t)$ in $L^2$, with $x(t) \in \mathbb C^d$ and $u(t) \in \mathbb C^m$, using as a basic tool a representation formula for its solution in terms of the initial condition, the control $u$, and some suitable matrix coefficients. When $\Lambda_1, \dotsc, \Lambda_N$ are commensurable, approximate and exact controllability are equivalent and can be characterized by a Kalman criterion. This paper focuses on providing characterizations of approximate and exact controllability without the commensurability assumption. In the case of two-dimensional systems with two delays, we obtain an explicit characterization of approximate and exact controllability in terms of the parameters of the problem. In the general setting, we prove that approximate controllability from zero to constant states is equivalent to approximate controllability in $L^2$. The corresponding result for exact controllability is true at least for two-dimensional systems with two delays. (10.5802/jep.112)
  DOI : 10.5802/jep.112