Publications

2021

• Mathematical and numerical analyses for the div-curl and div-curlcurl problems with a sign-changing coefficient
  
  • Ciarlet Patrick
  , 2021. We study the numerical approximation by edge finite elements of fields whose divergence and curl, or divergence and curl-curl, are prescribed in a bounded set $\Omega$ of $\mathbb{R}^3$, together with a boundary condition. Special attention is paid to solutions with low-regularity, in terms of the Sobolev scale $({\mathbf H}^{s}(\Omega))_{s>0}$. Among others, we consider an electromagnetic-like model including an interface between a classical medium and a metamaterial. In this setting the electric permittivity, and possibly the magnetic permeability, exhibit a sign-change at the interface. With the help of T-coercivity, we address the case of a model with one sign-changing coefficient, both for the model itself, and for its discrete version. Optimal error estimates are derived. Thanks to these results, we are also able to analyze the classical time-harmonic Maxwell equations, with one sign-changing coefficient.
• A DtN approach to the mathematical and numerical analysis in waveguides with periodic outlets at infinity
  
  • Fliss Sonia
  • Joly Patrick
  • Lescarret Vincent
  Pure and Applied Analysis, Mathematical Sciences Publishers, 2021. We consider the time harmonic scalar wave equation in junctions of several different periodic half-waveguides. In general this problem is not well posed. Several papers propose radiation conditions, i.e. the prescription of the behaviour of the solution at the infinities. This ensures uniqueness - except for a countable set of frequencies which correspond to the resonances- and yields existence when one is able to apply Fredholm alternative. This solution is called the outgoing solution. However, such radiation conditions are difficult to handle numerically. In this paper, we propose so-called transparent boundary conditions which enables us to characterize the outgoing solution. Moreover, the problem set in a bounded domain containing the junction with this transparent boundary conditions is of Fredholm type. These transparent boundary conditions are based on Dirichlet-to-Neumann operators whose construction is described in the paper. On contrary to the other approaches, the advantage of this approach is that a numerical method can be naturally derived in order to compute the outgoing solution. Numerical results illustrate and validate the method. (10.2140/paa.2021.3.487)
  DOI : 10.2140/paa.2021.3.487
• Imaging junctions of waveguides
  
  • Bourgeois Laurent
  • Fritsch Jean-François
  • Recoquillay Arnaud
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2021. In this paper we address the identification of defects by the Linear Sampling Method in half-waveguides which are related to each other by junctions. Firstly a waveguide which is characterized by an abrupt change of properties is considered, secondly the more difficult case of several half-waveguides related to each other by a junction of complex geometry. Our approach is illustrated by some two-dimensional numerical experiments. (10.3934/ipi.2020065)
  DOI : 10.3934/ipi.2020065
• Modèles homogénéisés enrichis en présence de bords : Analyse et traitement numérique
  
  • Beneteau Clément
  , 2021. Quand on s’intéresse à la propagation des ondes dans un milieu périodique à basse fréquence (i.e. la longueur d’onde est grande devant la période), il est possible de modéliser le milieu périodique par un milieu homogène équivalent ou effectif qui a les mêmes propriétés macroscopiques. C’est la théorie de l’homogénéisation qui justifie d’un point de vue mathématique ce procédé. Ce procédé est très séduisant car les calculs numériques sont beaucoup moins couteux (la petite structure périodique a disparu) et des calculs analytiques sont de nouveau possibles dans certaines configurations. Les ondes dans le milieu périodique et dans le milieu effectif sont très proches d’un point de vue macroscopique sauf en présence de bords ou d’interfaces.En effet, il est bien connu que le modèle homogénéisé est obtenu en négligeant les effets de bords et par conséquent il est beaucoup moins précis aux bords du milieu périodique. Quand les phénomènes intéressants apparaissent aux bords du milieu (comme la propagation des ondes plasmoniques à la surface des métamatériaux par exemple), il semble donc difficile de faire confiance au modèle effectif.En revenant sur le processus d’homogénéisation, nous proposons un modèle homogénéisé qui est plus riche aux niveaux des bords. Le modèle homogénéisé enrichi est aussi simple que le modèle homogénéisé classique loin des interfaces, seule les conditions aux bords changent et prennent mieux en compte les phénomènes. Nous appliquons ce modèle à une équation elliptique dans le cas de la géométrie simple du demi-plan avec des conditions de type Dirichlet ou Neumann. D’un point de vue numérique, en plus des problèmes de cellule classiques qui apparaissent en homogénéisation, des problèmes de bandes périodiques doivent également être résolus. Pour finir, nous appliquons ces résultats à l'homogénéisation de l'équation des ondes en temps long et en présence de bords.
• Propagation dans les guides d'ondes
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Lunéville Éric
  , 2021. On va s'intéresser dans ce cours à la résolution des équations de l'acoustique dans un guide fermé, c'est à dire un milieu cylindrique de section transverse bornée, en régime périodique établi. Dans le premier chapitre, nous nous intéressons à un guide parfait (sans défaut) : nous montrons alors que la propagation peut être décrite à l'aide de solutions particulières, à variables séparées, appelés modes. Dans les chapitres suivants, nous montrons comment étudier ou simuler l'effet d'un défaut ou d'une perturbation du guide sur un tel mode. On présentera en particulier des méthodes permettant de calculer par éléments finis le champ diffracté par le défaut : la difficulté concerne l'écriture de conditions aux limites non réfléchissantes sur les frontières artificielles du domaine de calcul. L'intérêt de la thématique de ce cours est double : D'une part, les guides d'ondes sont présents dans de nombreux domaines d'applications. Ils peuvent être naturels (la mer est un guide acoustique) ou fabriqués par l'homme (ligne co-axiale, plaque élastique etc...). La présence du défaut peut à son tour être accidentelle (fissure dans une plaque élastique) ou voulue (chambre d'expansion jouant le rôle de filtre dans un silencieux d'automobile). Pour un défaut non souhaité, il est intéressant de pouvoir le localiser en mesurant sa réponse à une onde incidente, c'est l'objectif du CND (Contrôle Non Destructif) par ultrasons. Pour une perturbation voulue, l'intérêt de la simulation est d'accéder à une évaluation précise de son effet. D'autre part, nous verrons que les guides d'ondes offrent un cadre assez simple (on utilisera beaucoup la séparation de variables en coordonnées cartésiennes) pour présenter et étudier des méthodes plus générales : en particulier, les techniques de conditions transparentes que nous présenterons (opérateurs DtN et couches PML) sont également utilisées pour des simulations dans des domaines de propagation qui ne sont pas des guides, et peuvent être infinis dans 2 ou 3 directions.
• Scattering of acoustic waves by a nonlinear resonant bubbly screen
  
  • Pham Kim
  • Mercier Jean-François
  • Fuster Daniel
  • Marigo Jean-Jacques
  • Maurel Agnès
  Journal of Fluid Mechanics, Cambridge University Press (CUP), 2021, 906, pp.A19. Some of the authors of this publication are also working on these related projects: PARIS code View project Homogenization of thin and thick microstructured materials View project (10.1017/jfm.2020.799)
  DOI : 10.1017/jfm.2020.799
• Solving unconstrained 0-1 polynomial programs through quadratic convex reformulation
  
  • Elloumi Sourour
  • Lambert Amélie
  • Lazare Arnaud
  Journal of Global Optimization, Springer Verlag, 2021, 80 (2), pp.231-248. We propose a solution approach for the problem (P) of minimizing an unconstrained binary polynomial optimization problem. We call this method PQCR (Polynomial Quadratic Convex Reformulation). The resolution is based on a 3-phase method. The first phase consists in reformulating (P) into a quadratic program (QP). For this, we recursively reduce the degree of (P) to two, by use of the standard substitution of the product of two variables by a new one. We then obtain a linearly constrained binary program. In the second phase, we rewrite the quadratic objective function into an equivalent and parametrized quadratic function using the equality x 2 i = x i and new valid quadratic equalities. Then, we focus on finding the best parameters to get a quadratic convex program which continuous relaxation's optimal value is maximized. For this, we build a semidefinite relaxation (SDP) of (QP). Then, we prove that the standard linearization inequalities, used for the quadratization step, are redundant in (SDP) in presence of the new quadratic equalities. Next, we deduce our optimal parameters from the dual optimal solution of (SDP). The third phase consists in solving (QP *), the optimal reformulated problem, with a standard solver. In particular, at each node of the branch-and-bound, the solver computes the optimal value of a continuous quadratic convex program. We present computational results on instances of the image restoration problem and of the low autocorrelation binary sequence problem. We compare PQCR with other convexification methods, and with the general solver Baron 17.4.1 [39]. We observe that most of the considered instances can be solved with our approach combined with the use of Cplex [24]. (10.1007/s10898-020-00972-2)
  DOI : 10.1007/s10898-020-00972-2
• General-purpose kernel regularization of boundary integral equations via density interpolation
  
  • Maltez Faria Luiz
  • Pérez-Arancibia Carlos
  • Bonnet Marc
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2021, 378, pp.113703. This paper presents a general high-order kernel regularization technique applicable to all four integral operators of Calder\'on calculus associated with linear elliptic PDEs in two and three spatial dimensions. Like previous density interpolation methods, the proposed technique relies on interpolating the density function around the kernel singularity in terms of solutions of the underlying homogeneous PDE, so as to recast singular and nearly singular integrals in terms of bounded (or more regular) integrands. We present here a simple interpolation strategy which, unlike previous approaches, does not entail explicit computation of high-order derivatives of the density function along the surface. Furthermore, the proposed approach is kernel- and dimension-independent in the sense that the sought density interpolant is constructed as a linear combination of point-source fields, given by the same {Green's function} used in the integral equation formulation, thus making the procedure applicable, in principle, to any PDE with known {Green's function}. For the sake of definiteness, we focus here on Nystr\"om methods for the (scalar) Laplace and Helmholtz equations and the (vector) elastostatic and time-harmonic elastodynamic equations. The method's accuracy, flexibility, efficiency, and compatibility with fast solvers are demonstrated by means of a variety of large-scale three-dimensional numerical examples. (10.1016/j.cma.2021.113703)
  DOI : 10.1016/j.cma.2021.113703
• BSDEs with no driving martingale, Markov processes and associated Pseudo Partial Differential Equations. Part II: Decoupled mild solutions and Examples.
  
  • Barrasso Adrien
  • Russo Francesco
  Journal of Theoretical Probability, Springer, 2021, 34, pp.1110–1148. Let $(\mathbb{P}^{s,x})_{(s,x)\in[0,T]\times E}$ be a family of probability measures, where $E$ is a Polish space,defined on the canonical probability space ${\mathbb D}([0,T],E)$ of $E$-valued cadlag functions. We suppose that a martingale problem with respect to a time-inhomogeneous generator $a$ is well-posed. We consider also an associated semilinear {\it Pseudo-PDE} with generator $a$ for which we introduce a notion of so called {\it decoupled mild} solution and study the equivalence with the notion of martingale solution introduced in a companion paper. We also investigate well-posedness for decoupled mild solutions and their relations with a special class of BSDEs without driving martingale. The notion of decoupled mild solution is a good candidate to replace the notion of viscosity solution which is not always suitable when the map $a$ is not a PDE operator. (10.1007/s10959-021-01092-7)
  DOI : 10.1007/s10959-021-01092-7
• Effective wave motion in periodic discontinua near spectral singularities at finite frequencies and wavenumbers
  
  • Guzina Bojan B
  • Bonnet Marc
  Wave Motion, Elsevier, 2021, 103, pp.102729. We consider the effective wave motion, at spectral singularities such as corners of the Brillouin zone and Dirac points, in periodic continua intercepted by compliant interfaces that pertain to e.g. masonry and fractured materials. We assume the Bloch-wave form of the scalar wave equation (describing anti-plane shear waves) as a point of departure, and we seek an asymptotic expansion about a reference point in the wavenumber-frequency space-deploying wavenumber separation as the perturbation parameter. Using the concept of broken Sobolev spaces to cater for the presence of kinematic discontinuities, we next define the "mean" wave motion via inner product between the Bloch wave and an eigenfunction (at specified wavenumber and frequency) for the unit cell of periodicity. With such projection-expansion approach, we obtain an effective field equation, for an arbitrary dispersion branch, near apexes of "wavenumber quadrants" featured by the first Brillouin zone. For completeness, we investigate asymptotic configurations featuring both (a) isolated, (b) repeated, and (c) nearby eigenvalues. In the case of repeated eigenvalues, we find that the "mean" wave motion is governed by a system of wave equations and Dirac equations, whose size is given by the eigenvalue multiplicity, and whose structure is determined by the participating eigenfunctions, the affiliated cell functions, and the direction of wavenumber perturbation. One of these structures is shown to describe the so-called Dirac points-apexes of locally conical dispersion surfaces-that are relevant to the generation of topologically protected waves. In situations featuring clusters of tightly spaced eigenvalues, the effective model is found to entail a Diraclike system of equations that generates "blunted" conical dispersion surfaces. We illustrate the analysis by numerical simulations for two periodic configurations in R 2 that showcase the asymptotic developments in terms of (i) wave dispersion, (ii) forced wave motion, and (iii) frequency-and wavenumber-dependent phonon behavior. (10.1016/j.wavemoti.2021.102729)
  DOI : 10.1016/j.wavemoti.2021.102729
• On a surprising instability result of Perfectly Matched Layers for Maxwell's equations in 3D media with diagonal anisotropy
  
  • Bécache Eliane
  • Fliss Sonia
  • Kachanovska Maryna
  • Kazakova Maria
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2021. The analysis of Cartesian Perfectly Matched Layers (PMLs) in the context of time-domain electromagnetic wave propagation in a 3D unbounded anisotropic homogeneous medium modelled by a diagonal dielectric tensor is presented. Contrary to the 3D scalar wave equation or 2D Maxwell's equations some diagonal anisotropies lead to the existence of backward waves giving rise to instabilities of the PMLs. Numerical experiments confirm the presented result. (10.5802/crmath.165)
  DOI : 10.5802/crmath.165
• Optimization of wireless sensor networks deployment with coverage and connectivity constraints
  
  • Elloumi Sourour
  • Hudry Olivier
  • Marie Estel
  • Martin Agathe
  • Plateau Agnès
  • Rovedakis Stephane
  Annals of Operations Research, Springer Verlag, 2021, 298 (1-2), pp.183-206. Wireless sensor networks have been widely deployed in the last decades to provide various services, like environmental monitoring or object tracking. Such a network is composed of a set of sensor nodes which are used to sense and transmit collected information to a base station. To achieve this goal, two properties have to be guaranteed: (i) the sensor nodes must be placed such that the whole environment of interest (represented by a set of targets) is covered, and (ii) every sensor node can transmit its data to the base station (through other sensor nodes). In this paper, we consider the Minimum Connected k-Coverage (MCkC) problem, where a positive integer k ≥ 1 defines the coverage multiplicity of the targets. We propose two mathematical programming formulations for the MCkC problem on square grid graphs and random graphs. We compare them to a recent model proposed by (Rebai et al 2015). We use a standard mixed integer linear programming solver to solve several instances with different formulations. In our results, we point out the quality of the LP-bound of each formulation as well as the total CPU time or the proportion of solved instances to optimality within a given CPU time. (10.1007/s10479-018-2943-7)
  DOI : 10.1007/s10479-018-2943-7
• On Weyl's type theorems and genericity of projective rigidity in sub-Riemannian Geometry
  
  • Jean Frédéric
  • Maslovskaya Sofya
  • Zelenko Igor
  Geometriae Dedicata, Springer Verlag, 2021, 213 (1), pp.295-314. H. Weyl in 1921 demonstrated that for a connected manifold of dimension greater than $1$, if two Riemannian metrics are conformal and have the same geodesics up to a reparametrization, then one metric is a constant scaling of the other one. In the present paper, we investigate the analogous property for sub-Riemannian metrics. In particular, we prove that the analogous statement, called the Weyl projective rigidity, holds either in real analytic category for all sub-Riemannian metrics on distributions with a specific property of their complex abnormal extremals, called minimal order, or in smooth category for all distributions such that all complex abnormal extremals of their nilpotent approximations are of minimal order. This also shows, in real analytic category, the genericity of distributions for which all sub-Riemannian metrics are Weyl projectively rigid and genericity of Weyl projectively rigid sub-Riemannian metrics on a given bracket generating distributions. Finally, this allows us to get analogous genericity results for projective rigidity of sub-Riemannian metrics, i.e.when the only sub-Riemannian metric having the same sub-Riemannian geodesics, up to a reparametrization, with a given one, is a constant scaling of this given one. This is the improvement of our results on the genericity of weaker rigidity properties proved in recent paper arXiv:1801.04257[math.DG]. (10.1007/s10711-020-00581-z)
  DOI : 10.1007/s10711-020-00581-z
• Martingale driven BSDEs, PDEs and other related deterministic problems
  
  • Barrasso Adrien
  • Russo Francesco
  Stochastic Processes and their Applications, Elsevier, 2021, 133, pp.193-228. We focus on a class of BSDEs driven by a cadlag martingale and corresponding Markov type BSDE which arise when the randomness of the driver appears through a Markov process. To those BSDEs we associate a deterministic problem which, when the Markov process is a Brownian diffusion, is nothing else but a parabolic type PDE. The solution of the deterministic problem is intended as decoupled mild solution, and it is formulated with the help of a time-inhomogeneous semigroup. (10.1016/j.spa.2020.11.007)
  DOI : 10.1016/j.spa.2020.11.007
• Global optimization approach for the ascent problem of multi-stage launchers
  
  • Bokanowski Olivier
  • Bourgeois Eric
  • Desilles Anna
  • Zidani Hasnaa
  , 2021, pp.1--42. This paper deals with a problem of trajectory optimization of the flight phases of a three-stage launcher. The aim of this optimization problem is to minimize the consumption of ergols that is need to steer the launcher from the Earth to the GEO. Here we use a global optimization procedure based on Hamilton-Jacobi-Bellman approach and consider a complete model including the transfer from the GTO to the GEO orbit. The Hamilton-Jacobi approach proposed here takes also into account parametric optimisation that appears in the flight phases. The work presented in this paper has been performed in the frame of CNES Launchers' Research and Technology program (10.1007/978-3-030-55240-4_1)
  DOI : 10.1007/978-3-030-55240-4_1
• Local transparent boundary conditions for wave propagation in fractal trees (I). Method and numerical implementation
  
  • Joly Patrick
  • Kachanovska Maryna
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2021. This work is dedicated to the construction and analysis of high-order transparentboundary conditions for the weighted wave equation on a fractal tree, which models sound propaga-tion inside human lungs. This article follows the works [9, 6], aimed at the analysis and numerical treatment of the model, as well as the construction of low-order and exact discrete boundary conditions. The method suggested in the present work is based on the truncation of the meromorphicseries that represents the symbol of the Dirichlet-to-Neumann operator, in the spirit of the absorbingboundary conditions of B. Engquist and A. Majda. We analyze its stability and convergence, as wellas present computational aspects of the method. Numerical results confirm theoretical findings (10.1137/20M1362334)
  DOI : 10.1137/20M1362334
• An automatic PML for acoustic finite element simulations in convex domains of general shape
  
  • Bériot Hadrien
  • Modave Axel
  International Journal for Numerical Methods in Engineering, Wiley, 2021, 122 (5), pp.1239-1261. This article addresses the efficient finite element solution of exterior acoustic problems with truncated computational domains surrounded by perfectly matched layers (PMLs). The PML is a popular nonreflecting technique that combines accuracy, computational efficiency, and geometric flexibility. Unfortunately, the effective implementation of the PML for convex domains of general shape is tricky because of the geometric parameters that are required to define the PML medium. In this work, a comprehensive implementation strategy is proposed. This approach, which we call the automatically matched layer (AML) implementation, is versatile and fully automatic for the end‐user. With the AML approach, the mesh of the layer is extruded, the required geometric parameters are automatically obtained during the extrusion step, and the practical implementation relies on a simple modification of the Jacobian matrix in the elementwise integrals. The AML implementation is validated and compared with other implementation strategies using numerical benchmarks in two and three dimensions, considering computational domains with regular and nonregular boundaries. A three‐dimensional application with a generally shaped domain generated using a convex hull is proposed to illustrate the interest of the AML approach for realistic industrial cases. (10.1002/nme.6560)
  DOI : 10.1002/nme.6560
• Stability and Convergence Analysis of Time-domain Perfectly Matched Layers for The Wave Equation in Waveguides
  
  • Bécache Eliane
  • Kachanovska Maryna
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2021. This work is dedicated to the proof of stability and convergence of the Bérenger's perfectly matched layers in the waveguides for an arbitrary L ∞ damping function. The proof relies on the Laplace domain techniques and an explicit representation of the solution to the PML problem in the waveguide. A bound for the PML error that depends on the absorption parameter and the length of the PML is presented. Numerical experiments confirm the theoretical findings. (10.1137/20M1330543)
  DOI : 10.1137/20M1330543
• Homogenization of Maxwell's equations and related scalar problems with sign-changing coefficients
  
  • Bunoiu Renata
  • Chesnel Lucas
  • Ramdani Karim
  • Rihani Mahran
  Annales de la Faculté des Sciences de Toulouse. Mathématiques., Université Paul Sabatier _ Cellule Mathdoc, 2021, 30 (5), pp.1075-1119. In this work, we are interested in the homogenization of time-harmonic Maxwell's equations in a composite medium with periodically distributed small inclusions of a negative material. Here a negative material is a material modelled by negative permittivity and permeability. Due to the sign-changing coefficients in the equations, it is not straightforward to obtain uniform energy estimates to apply the usual homogenization techniques. The goal of this article is to explain how to proceed in this context. The analysis of Maxwell's equations is based on a precise study of two associated scalar problems: one involving the sign-changing permittivity with Dirichlet boundary conditions, another involving the sign-changing permeability with Neumann boundary conditions. For both problems, we obtain a criterion on the physical parameters ensuring uniform invertibility of the corresponding operators as the size of the inclusions tends to zero. In the process, we explain the link existing with the so-called Neumann-Poincaré operator, complementing the existing literature on this topic. Then we use the results obtained for the scalar problems to derive uniform energy estimates for Maxwell's system. At this stage, an additional difficulty comes from the fact that Maxwell's equations are also sign-indefinite due to the term involving the frequency. To cope with it, we establish some sort of uniform compactness result. (10.5802/afst.1694)
  DOI : 10.5802/afst.1694
• Weak Input to state estimates for 2D damped wave equations with localized and non-linear damping
  
  • Kafnemer Meryem
  • Mebkhout Benmiloud
  • Chitour Yacine
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2021. In this paper, we study input-to-state (ISS) issues for damped wave equations with Dirichlet boundary conditions on a bounded domain of dimension two. The damping term is assumed to be non-linear and localized to an open subset of the domain. In a first step, we handle the undisturbed case as an extension of a previous work, where stability results are given with a damping term active on the full domain. Then, we address the case with disturbances and provide input-to-state types of results. (10.1137/20M1337909)
  DOI : 10.1137/20M1337909
• Optimal slip velocities of micro-swimmers with arbitrary axisymmetric shapes
  
  • Guo Hanliang
  • Zhu Hai
  • Liu Ruowen
  • Bonnet Marc
  • Veerapaneni Shravan
  Journal of Fluid Mechanics, Cambridge University Press (CUP), 2021, 910, pp.A26. This article presents a computational approach for determining the optimal slip velocities on any given shape of an axisymmetric micro-swimmer suspended in a viscous fluid. The objective is to minimize the power loss to maintain a target swimming speed, or equivalently to maximize the efficiency of the micro-swimmer. Owing to the linearity of the Stokes equations governing the fluid motion, we show that this PDE-constrained optimization problem reduces to a simpler quadratic optimization problem, whose solution is found using a high-order accurate boundary integral method. We consider various families of shapes parameterized by the reduced volume and compute their swimming efficiency. {Among those, prolate spheroids were found to be the most efficient micro-swimmer shapes for a given reduced volume. We propose a simple shape-based scalar metric that can determine whether the optimal slip on a given shape makes it a pusher, a puller, or a neutral swimmer.} (10.1017/jfm.2020.969)
  DOI : 10.1017/jfm.2020.969
• Relationship Between Maximum Principle and Dynamic Programming in presence of Intermediate and Final State Constraints
  
  • Bokanowski Olivier
  • Desilles Anna
  • Zidani Hasnaa
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2021. In this paper, we consider a class of optimal control problems governed by a differential system. We analyze the sensitivity relations satisfied by the co-state arc of the Pontryagin maximum principle and the value function that associates the optimal value of the control problem to the initial time and state. Such a relationship has been already investigated for state-constrained problems under some controllability assumptions to guarantee Lipschitz regularity property of the value function. Here, we consider the case with intermediate and final state constraints, without any controllability assumption on the system, and without Lipschitz regularity of the value function. Because of this lack of regularity, the sensitivity relations cannot be expressed with the sub-differentials of the value function. This work shows that the constrained problem can be reformulated with an auxiliary value function which is more regular and suitable to express the sensitivity of the adjoint arc of the original state-constrained control problem along an optimal trajectory. Furthermore, our analysis covers the case of normal optimal solutions, and abnormal solutions as well. (10.1051/cocv/2021084)
  DOI : 10.1051/cocv/2021084
• Experimental and theoretical observations on DDT in smooth narrow channels
  
  • Melguizo-Gavilanes J.
  • Ballossier Yves
  • Maltez Faria Luiz
  Proceedings of the Combustion Institute, Elsevier, 2021, 38 (3). A combined experimental and theoretical study of deflagration-to-detonation transition (DDT) in smooth narrow channels is presented. Some of the distinguishing features characterizing the late stages of DDT are shown to be qualitatively captured by a simple one-dimensional scalar equation. Inspection of the structure and stability of the traveling wave solutions found in the model, and comparison with experimental observations, suggest a possible mechanism responsible for front acceleration and transition to detonation. (10.1016/j.proci.2020.07.142)
  DOI : 10.1016/j.proci.2020.07.142
• Variational Methods for Acoustic Radiation in a Duct with a Shear Flow and an Absorbing Boundary
  
  • Mercier Jean-François
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2021, 81 (6), pp.2658-2683. The well-posedness of the acoustic radiation in a 2D duct in presence of both a shear flow and an absorbing wall described by the Myers boundary condition is studied thanks to variational methods. Without flow the problem is found well-posed for any impedance value. The presence of a flow complicates the results. With a uniform flow the problem is proven to be always of the Fredholm type but is found well-posed only when considering a dissipative radiation problem. With a general shear flow, the Fredholm property is recovered for a weak enough shear and the dissipative radiation problem requires to introduce extra conditions to be well-posed: enough dissipation, a large enough frequency and non-intuitive conditions on the impedance value. (10.1137/20M1384026)
  DOI : 10.1137/20M1384026
• ROUGH PATHS AND REGULARIZATION
  
  • Gomes André O
  • Ohashi Alberto
  • Russo Francesco
  • Teixeira Alan
  Journal of Stochastic Analysis, Louisiana State University, 2021, 2 (4), pp.1-21. Calculus via regularizations and rough paths are two methods to approach stochastic integration and calculus close to pathwise calculus. The origin of rough paths theory is purely deterministic, calculus via regularization is based on deterministic techniques but there is still a probability in the background. The goal of this paper is to establish a connection between stochastically controlled-type processes, a concept reminiscent from rough paths theory, and the so-called weak Dirichlet processes. As a by-product, we present the connection between rough and Stratonovich integrals for càdlàg weak Dirichlet processes integrands and continuous semimartingales integrators. (10.31390/josa.2.4.01)
  DOI : 10.31390/josa.2.4.01