Publications

2021

• 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
• 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
• 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
• 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
• 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
• 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
• 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
• Optimal Ciliary Locomotion of Axisymmetric Microswimmers
  
  • Guo Hanliang
  • Zhu Hai
  • Liu Ruowen
  • Bonnet Marc
  • Veerapaneni Shravan
  Journal of Fluid Mechanics, Cambridge University Press (CUP), 2021, 927, pp.A22. Many biological microswimmers locomote by periodically beating the densely-packed cilia on their cell surface in a wave-like fashion. While the swimming mechanisms of ciliated microswimmers have been extensively studied both from the analytical and the numerical point of view, the optimization of the ciliary motion of microswimmers has received limited attention, especially for non-spherical shapes. In this paper, using an envelope model for the microswimmer, we numerically optimize the ciliary motion of a ciliate with an arbitrary axisymmetric shape. The forward solutions are found using a fast boundary integral method, and the efficiency sensitivities are derived using an adjoint-based method. Our results show that a prolate microswimmer with a 2:1 aspect ratio shares similar optimal ciliary motion as the spherical microswimmer, yet the swimming efficiency can increase two-fold. More interestingly, the optimal ciliary motion of a concave microswimmer can be qualitatively different from that of the spherical microswimmer, and adding a constraint to the ciliary length is found to improve, on average, the efficiency for such swimmers. (10.1017/jfm.2021.744)
  DOI : 10.1017/jfm.2021.744
• 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