Publications

2018

• Optimal Control of Infinite Dimensional Bilinear Systems: Application to the Heat and Wave Equations
  
  • Aronna Maria Soledad
  • Bonnans Joseph Fréderic
  • Kröner Axel
  Mathematical Programming, Springer Verlag, 2018, 168 (1-2), pp.717-757. In this paper we consider second order optimality conditions for a bilinear optimal control problem governed by a strongly continuous semigroup operator, the control entering linearly in the cost function. We derive first and second order optimality conditions, taking advantage of the Goh transform. We then apply the results to the heat and wave equations.
• Linearized Navier-Stokes Equations for Aeroacoustics using Stabilized Finite Elements: Boundary Conditions and Industrial Application to Aft-Fan Noise Propagation
  
  • Bissuel Aloïs
  • Allaire Grégoire
  • Daumas Laurent
  • Barré Sébastien
  • Rey Floriane
  Computers and Fluids, Elsevier, 2018. In this paper, a numerical method for solving the linearized Navier-Stokes equations is presented for aeroacoustic sound propagation problem. The Navier-Stokes equations are linearized in the frequency domain. The fan noise of jet engine is emitted nearly selectively at certain frequencies, which depend on the rotation velocity of the fan. A frequency domain approach is highly suitable for this kind of problem, instead of a costly time-dependent simulation which can handle a large range of frequencies depending on the time step and the mesh. The calculations presented here were all made using Aether, a Navier-Stokes code which uses finite elements stabilized with SUPG (Streamline Upwind Galerkin). Automatic code differentiation was used to linearize this code. Entropy variables bring interesting mathematical properties to the numerical scheme, but also prevent the easy implementation of boundary conditions. For instance, the pressure is a non-linear combination of the entropy variables. Imposing a pressure variation needs a linearization of this relation which is detailed herein. The performance of different types of boundary conditions used to impose the acoustic pressure variation inside the engine is studied in detail. Finally, a very surprising effect of the SUPG scheme was to transform a homogeneous Dirichlet boundary condition on all variables to a transparent one which is able to let only outgoing waves pass * Dassault Aviation 78, quai Marcel 1 through with no incoming wave. A one-dimensional toy model is given to explain how SUPG brings about this transformation. The last part of the article is dedicated to an industrial test case. The geometry of a model turbine from the Clean Sky European project was used for sound propagation of the fan exhaust noise of a jet engine. Computations on several modes with increasing complexities were done and the results compared to a boundary element method which served as a reference when no mean flow is present. Results of a computation with a mean flow are shown.
• Gauge-reversing maps on cones, and Hilbert and Thompson isometries
  
  • Walsh Cormac
  Geometry and Topology, Mathematical Sciences Publishers, 2018, 22 (1), pp.55-104. We show that a cone admits a gauge-reversing map if and only if it is a symmetric cone. We use this to prove that every isometry of a Hilbert geometry is a collineation unless the Hilbert geometry is the projective space of a non-Lorentzian symmetric cone, in which case the collineation group is of index two in the isometry group. We also determine the isometry group of the Thompson geometry on a cone. (10.2140/gt.2018.22.55)
  DOI : 10.2140/gt.2018.22.55
• On the Whitney extension property for continuously differentiable horizontal curves in sub-Riemannian manifolds
  
  • Sacchelli Ludovic
  • Sigalotti Mario
  Calculus of Variations and Partial Differential Equations, Springer Verlag, 2018. In this article we study the validity of the Whitney $C^1$ extension property for horizontal curves in sub-Riemannian manifolds endowed with 1-jets that satisfy a first-order Taylor expansion compatibility condition. We first consider the equiregular case, where we show that the extension property holds true whenever a suitable non-singularity property holds for the input-output maps on the Carnot groups obtained by nilpotent approximation. We then discuss the case of sub-Riemannian manifolds with singular points and we show that all step-2 manifolds satisfy the $C^1$ extension property. We conclude by showing that the $C^1$ extension property implies a Lusin-like approximation theorem for horizontal curves on sub-Riemannian manifolds.