Publications

2022

• Optimal planetary landing with pointing and glide-slope constraints
  
  • Leparoux Clara
  • Hérissé Bruno
  • Jean Frédéric
  , 2022, pp.4357-4362. This paper studies a vertical powered descent problem in the context of planetary landing, considering glide-slope and thrust pointing constraints and minimizing any final cost. After stating the Max-Min-Max or Max-Singular-Max form of the optimal control deduced from the Pontryagin Maximum Principle, it theoretically analyzes the optimal trajectory for a more specific problem formulation to show that there can be at most one contact or boundary interval with the state constraint on each Max or Min arc. (10.1109/CDC51059.2022.9992735)
  DOI : 10.1109/CDC51059.2022.9992735
• Influence of chemistry on the steady solutions of hydrogen gaseous detonations with friction losses
  
  • Veiga-Lopez Fernando
  • Maltez Faria Luiz
  • Melguizo-Gavilanes J.
  Combustion and Flame, Elsevier, 2022, 240, pp.112050. The problem of the steady propagation of detonation waves with friction losses is revisited including detailed kinetics. The derived formulation is used to study the influence of chemical modeling on the steady solutions and reaction zone structures obtained for stoichiometric hydrogen-oxygen. Detonation velocity-friction coefficient (D − c f) curves, pressure, temperature, Mach number, thermicity and species profiles are used for that purpose. Results show that both simplified kinetic schemes considered (i.e., one-step and three-step chainbranching), fitted using standard methodologies, failed to quantitatively capture the critical c f values obtained with detailed kinetics; moreover one-step Arrhenius chemistry also exhibits qualitative differences for D/D CJ ≤ 0.55 due to an overestimation of the chemical time in this regime. An alternative fitting methodology for simplified kinetics is proposed using detailed chemistry D − c f curves as a target rather than constant volume delay times and ideal Zel'dovich-von Neumann-Döring profiles; this method is in principle more representative to study non-ideal detonation propagation. The sensitivity of the predicted critical c f value, c f,crit , to the detailed mechanisms routinely used to model hydrogen oxidation was also assessed; significant differences were found, mainly driven by the consumption/creation rate of the HO 2 radical pool at low postshock temperature. (10.1016/j.combustflame.2022.112050)
  DOI : 10.1016/j.combustflame.2022.112050
• Time-Varying Optimization of Networked Systems With Human Preferences
  
  • Ospina Ana
  • Simonetto Andrea
  • Dall'Anese Emiliano
  IEEE Transactions on Control of Network Systems, IEEE, 2022, pp.1-12. (10.1109/TCNS.2022.3203467)
  DOI : 10.1109/TCNS.2022.3203467
• Backward Stochastic Differential Equations with no driving martingale, Markov processes and associated Pseudo Partial Differential Equations
  
  • Barrasso Adrien
  • Russo Francesco
  Journal of Stochastic Analysis, Louisiana State University, 2022, 3 (1). We discuss a class of Backward Stochastic Differential Equations (BSDEs) with no driving martingale. When the randomness of the driver depends on a general Markov process $X$, those BSDEs are denominated Markovian BSDEs and can be associated to a deterministic problem, called Pseudo-PDE which constitute the natural generalization of a parabolic semilinear PDE which naturally appears when the underlying filtration is Brownian. We consider two aspects of well-posedness for the Pseudo-PDEs: "classical" and "martingale" solutions. (10.31390/josa.3.1.03)
  DOI : 10.31390/josa.3.1.03
• Limiting amplitude principle and resonances in plasmonic structures with corners: numerical investigation
  
  • Carvalho Camille
  • Ciarlet Patrick
  • Scheid Claire
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2022, 388, pp.114207. The limiting amplitude principle states that the response of a scatterer to a harmonic light excitation is asymptotically harmonic with the same pulsation. Depending on the geometry and nature of the scatterer, there might or might not be an established theoretical proof validating this principle. In this paper, we investigate a case where the theory is missing: we consider a two-dimensional dispersive Drude structure with corners. In the non lossy case, it is well known that looking for harmonic solutions leads to an ill-posed problem for a specific range of critical pulsations, characterized by the metal’s properties and the aperture of the corners. Ill-posedness is then due to highly oscillatory resonances at the corners called black-hole waves. However, a time-domain formulation with a harmonic excitation is always mathematically valid. Based on this observation, we conjecture that the limiting amplitude principle might not hold for all pulsations. Using a time-domain setting, we propose a systematic numerical approach that allows to give numerical evidences of the latter conjecture, and find clear signature of the critical pulsa- tions. Furthermore, we connect our results to the underlying physical plasmonic resonances that occur in the lossy physical metallic case. (10.1016/j.cma.2021.114207)
  DOI : 10.1016/j.cma.2021.114207
• On some path-dependent SDEs involving distributional drifts
  
  • Ohashi Alberto
  • Russo Francesco
  • Teixeira Alan
  Modern Stochastics: Theory and Applications, VTEX, 2022, 9 (1), pp.65-87. In this paper, we study (strong and weak) existence and uniqueness of a class of non-Markovian SDEs whose drift contains the derivative in the sense of distributions of a continuous function. (10.15559/21-VMSTA197)
  DOI : 10.15559/21-VMSTA197
• On the Half-Space Matching Method for Real Wavenumber
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chandler-Wilde Simon N
  • Fliss Sonia
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2022, 82 (4). The Half-Space Matching (HSM) method has recently been developed as a new method for the solution of 2D scattering problems with complex backgrounds, providing an alternative to Perfectly Matched Layers (PML) or other artificial boundary conditions. Based on half-plane representations for the solution, the scattering problem is rewritten as a system coupling (1) a standard finite element discretisation localised around the scatterer and (2) integral equations whose unknowns are traces of the solution on the boundaries of a finite number of overlapping half-planes contained in the domain. While satisfactory numerical results have been obtained for real wavenumbers, well-posedness and equivalence of this HSM formulation to the original scattering problem have been established only for complex wavenumbers. In the present paper we show, in the case of a homogeneous background, that the HSM formulation is equivalent to the original scattering problem also for real wavenumbers, and so is well-posed, provided the traces satisfy radiation conditions at infinity analogous to the standard Sommerfeld radiation condition. As a key component of our argument we show that, if the trace on the boundary of a half-plane satisfies our new radiation condition, then the corresponding solution to the half-plane Dirichlet problem satisfies the Sommerfeld radiation condition in a slightly smaller half-plane. We expect that this last result will be of independent interest, in particular in studies of rough surface scattering. (10.1137/21M1459216)
  DOI : 10.1137/21M1459216
• The Complex-Scaled Half-Space Matching Method
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chandler-Wilde Simon N.
  • Fliss Sonia
  • Hazard Christophe
  • Perfekt Karl-Mikael
  • Tjandrawidjaja Yohanes
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2022, 54 (1), pp.512-557. The Half-Space Matching (HSM) method has recently been developed as a new method for the solution of 2D scattering problems with complex backgrounds, providing an alternative to Perfectly Matched Layers (PML) or other artificial boundary conditions. Based on half-plane representations for the solution, the scattering problem is rewritten as a system of integral equations in which the unknowns are restrictions of the solution to the boundaries of a finite number of overlapping half-planes contained in the domain: this integral equation system is coupled to a standard finite element discretisation localised around the scatterer. While satisfactory numerical results have been obtained for real wavenumbers, wellposedness and equivalence to the original scattering problem have been established only for complex wavenumbers. In the present paper, by combining the HSM framework with a complex-scaling technique, we provide a new formulation for real wavenumbers which is provably well-posed and has the attraction for computation that the complex-scaled solutions of the integral equation system decay exponentially at infinity. The analysis requires the study of double-layer potential integral operators on intersecting infinite lines, and their analytic continuations. The effectiveness of the method is validated by preliminary numerical results. (10.1137/20M1387122)
  DOI : 10.1137/20M1387122
• A mathematical study of a hyperbolic metamaterial in free space
  
  • Ciarlet Patrick
  • Kachanovska Maryna
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2022, 54 (2), pp.2216-2250. Wave propagation in hyperbolic metamaterials is described by the Maxwell equations with a frequency dependent tensor of dielectric permittivity, whose eigenvalues are of different signs. In this case the problem becomes hyperbolic (Klein-Gordon equation) for a certain range of frequencies. The principal theoretical and numerical difficulty comes from the fact that this hyperbolic equation is posed in a free space, without initial conditions provided. The subject of the work is the theoretical justification of this problem. In particular, this includes the construction of a radiation condition, a well-posedness result, a limiting absorption principle and regularity estimates on the solution. (10.1137/21M1404223)
  DOI : 10.1137/21M1404223
• Fokker-Planck equations with terminal condition and related McKean probabilistic representation
  
  • Izydorczyk Lucas
  • Oudjane Nadia
  • Russo Francesco
  • Tessitore Gianmario
  Nonlinear Differential Equations and Applications, Springer Verlag, 2022, volume 29 (10). Usually Fokker-Planck type partial differential equations (PDEs) are well-posed if the initial condition is specified. In this paper, alternatively, we consider the inverse problem which consists in prescribing final data: in particular we give sufficient conditions for existence and uniqueness. In the second part of the paper we provide a probabilistic representation of those PDEs in the form a solution of a McKean type equation corresponding to the time-reversal dynamics of a diffusion process. (10.1007/s00030-021-00736-1)
  DOI : 10.1007/s00030-021-00736-1
• Local transparent boundary conditions for wave propagation in fractal trees (ii): error and complexity analysis
  
  • Joly Patrick
  • Kachanovska Maryna
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2022, 60 (2). This work is dedicated to a refined error analysis of the high-order transparent boundary conditions introduced in the companion work [8] for the weighted wave equation on a fractal tree. The construction of such boundary conditions relies on truncating the meromorphic series that represents the symbol of the Dirichlet-to-Neumann operator. The error induced by the truncation depends on the behaviour of the eigenvalues and the eigenfunctions of the weighted Laplacian on a self-similar metric tree. In this work we quantify this error by computing asymptotics of the eigenvalues and bounds for Neumann traces of the eigenfunctions. We prove the sharpness of the obtained bounds for a class of self-similar trees. (10.1137/20M1357524)
  DOI : 10.1137/20M1357524
• A Mayer optimal control problem on Wasserstein spaces over Riemannian manifolds
  
  • Jean Frédéric
  • Jerhaoui Othmane
  • Zidani Hasnaa
  , 2022, 55 (16), pp.44-49. (10.1016/j.ifacol.2022.08.079)
  DOI : 10.1016/j.ifacol.2022.08.079
• Gâteaux type path-dependent PDEs and BSDEs with Gaussian forward processes
  
  • Barrasso Adrien
  • Russo Francesco
  Stochastics and Dynamics, World Scientific Publishing, 2022, 22, pp.2250007,. We are interested in path-dependent semilinear PDEs, where the derivatives are of Gâteaux type in specific directions k and b, being the kernel functions of a Volterra Gaussian process X. Under some conditions on k, b and the coefficients of the PDE, we prove existence and uniqueness of a decoupled mild solution, a notion introduced in a previous paper by the authors. We also show that the solution of the PDE can be represented through BSDEs where the forward (underlying) process is X. (10.1142/S0219493722500071)
  DOI : 10.1142/S0219493722500071
• Relaxed-Inertial Proximal Point Type Algorithms for Quasiconvex Minimization
  
  • Grad Sorin-Mihai
  • Lara Felipe
  • Marcavillaca Raul Tintaya
  Journal of Global Optimization, Springer Verlag, 2022. We propose a relaxed-inertial proximal point type algorithm for solving optimization problems consisting in minimizing strongly quasiconvex functions whose variables lie in finitely dimensional linear subspaces. A relaxed version of the method where the constraint set is only closed and convex is also discussed, and so is the case of a quasiconvex objective function. Numerical experiments illustrate the theoretical results.
• Optimistic Planning Algorithms For State-Constrained Optimal Control Problems
  
  • Bokanowski Olivier
  • Gammoudi Nidhal
  • Zidani Hasnaa
  Computers & Mathematics with Applications, Elsevier, 2022, 109 (1), pp.158-179. In this work, we study optimistic planning methods to solve some state-constrained optimal control problems in finite horizon. While classical methods for calculating the value function are generally based on a discretization in the state space, optimistic planning algorithms have the advantage of using adaptive discretization in the control space. These approaches are therefore very suitable for control problems where the dimension of the control variable is low and allow to deal with problems where the dimension of the state space can be very high. Our algorithms also have the advantage of providing, for given computing resources, the best control strategy whose performance is as close as possible to optimality while its corresponding trajectory comply with the state constraints up to a given accuracy. (10.1016/j.camwa.2022.01.016)
  DOI : 10.1016/j.camwa.2022.01.016
• Distributed Personalized Gradient Tracking with Convex Parametric Models
  
  • Notarnicola Ivano
  • Simonetto Andrea
  • Farina Francesco
  • Notarstefano Giuseppe
  IEEE Transactions on Automatic Control, Institute of Electrical and Electronics Engineers, 2022, pp.1-1. (10.1109/TAC.2022.3147007)
  DOI : 10.1109/TAC.2022.3147007
• The Morozov's principle applied to data assimilation problems
  
  • Bourgeois Laurent
  • Dardé Jérémi
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2022. This paper is focused on the Morozov's principle applied to an abstract data assimilation framework, with particular attention to three simple examples: the data assimilation problem for the Laplace equation, the Cauchy problem for the Laplace equation and the data assimilation problem for the heat equation. Those ill-posed problems are regularized with the help of a mixed type formulation which is proved to be equivalent to a Tikhonov regularization applied to a well-chosen operator. The main issue is that such operator may not have a dense range, which makes it necessary to extend well-known results related to the Morozov's choice of the regularization parameter to that unusual situation. The solution which satisfies the Morozov's principle is computed with the help of the duality in optimization, possibly by forcing the solution to satisfy given a priori constraints. Some numerical results in two dimensions are proposed in the case of the data assimilation problem for the Laplace equation. (10.1051/m2an/2022061)
  DOI : 10.1051/m2an/2022061
• Structure of optimal control for planetary landing with control and state constraints
  
  • Leparoux Clara
  • Hérissé Bruno
  • Jean Frédéric
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2022, 28. This paper studies a vertical powered descent problem in the context of planetary landing, considering glide-slope and thrust pointing constraints and minimizing any final cost. In a first time, it proves the Max-Min-Max or Max-Singular-Max form of the optimal control using the Pontryagin Maximum Principle, and it extends this result to a problem formulation considering the effect of an atmosphere. It also shows that the singular structure does not appear in generic cases. In a second time, it theoretically analyzes the optimal trajectory for a more specific problem formulation to show that there can be at most one contact or boundary interval with the state constraint on each Max or Min arc. (10.1051/cocv/2022065)
  DOI : 10.1051/cocv/2022065
• On the approximation of electromagnetic fields by edge finite elements. Part 4: analysis of the model with one sign-changing coefficient
  
  • Ciarlet Patrick
  Numerische Mathematik, Springer Verlag, 2022, 152, pp.223-257. In electromagnetism, in the presence of a negative material surrounded by a classical material, the electric permittivity, and possibly the magnetic permeability, can exhibit a sign-change at the interface. In this setting, the study of electromagnetic phenomena is a challenging topic. We focus on the time-harmonic Maxwell equations in a bounded set $\Omega$ of ${\mathbb R}^3$, and more precisely on the numerical approximation of the electromagnetic fields by edge finite elements. Special attention is paid to low-regularity solutions, in terms of the Sobolev scale $({\boldsymbol{H}}^{\mathtt{s}}(\Omega))_{\mathtt{s}>0}$. With the help of T-coercivity, we address the case of one sign-changing coefficient, both for the model itself, and for its discrete version. Optimal a priori error estimates are derived. (10.1007/s00211-022-01315-x)
  DOI : 10.1007/s00211-022-01315-x
• Kernel representation of Kalman observer and associated H-matrix based discretization
  
  • Aussal Matthieu
  • Moireau Philippe
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2022, 28, pp.78. In deterministic estimation, applying a Kalman filter to a dynamical model based on partial differential equations is theoretically seducing but solving the associated Riccati equation leads to a so-called curse of dimensionality for its numerical implementation. In this work, we propose to entirely revisit the theory of Kalman filters for parabolic problems where additional regularity results proves that the Riccati equation solution belongs to the class of Hilbert-Schmidt operators. The regularity of the associated kernel then allows to proceed to the numerical analysis of the Kalman full space-time discretization in adapted norms, hence justifying the implementation of the related Kalman filter numerical algorithm with H-matrices typically developed for integral equations discretization. (10.1051/cocv/2022071)
  DOI : 10.1051/cocv/2022071