Section: New Results
Mathematical analysis in geometrical optimization
Fast algorithm for Maxwell 3D harmonic solution via optimal control approach
Participants : Luigi Manca, Jean-Paul Zolésio.
Using the pseudo differential boundary operator T, “Dirichlet to Neumann” for any outgoing radiating solution of the Maxwell equation in the exterior of a bounded domain D with boundary S, we build an optimal control problem whose solution is the harmonic regime in the domain D. The optimal control is part of the initial data for the time depending solution. Under periodic excitation (with compact support in D) and lateral Neumann boundary condition on S, the operator T is involved. The optimal synthesis involves the time backward adjoint state which is captured by the Ricatti solution. The mathematical proof of the « device » requires sharp regularity analysis on the non homgeneous Neuman problem associated in D with the time depending 3D Maxwell system [60] [32] .
Optimal geometry in radar device
Participants : Xavier Hachair, Jean-Paul Zolésio.
This is an confidential approach for the conception of a part of radar device. It involves a specific geometrical optimization.
Hidden sharp regularity and shape derivative in wave and hyperbolic systems
Participants : Michel Delfour, Jean-Paul Zolésio.
After ICIAM 1995 in Hamburg ( and several papers) we introduced the so-called “extractor technique” which permit to recover the hidden regularity results in wave equation Under Dirichlet Boundary Condition. These results were a kind of quantified Version of results derived by I.Lasiecka and R. Triggiani using some multiplicator techniques and were power full enough to derive shape dérivative Under weak regularity . Nevertheless this technique failed for the Neumann like boundary conditions. We introduce theew concept of “Pseudo-differential” technique which recentely dropped this limitation. So we develop new sharp regularity results leading for shape dérivative existence for wave Under eumann boundary data in the space of finité energy on the boundary. The intrinsic character of the pseudo extractor permits toextend easily the results to the important situation of free time depending elastic shell equations [27] .
Non cylindrical dynamical system
Participant : Jean-Paul Zolésio.
Optimal control theory is classicaly based on the assumption that the problem to be controled has solutions and is well posed when the control parameter describes a whole set (say a closed convex set) of some functional linear space. Concerning moving domains in classical heat or wave equations with usual boundary conditions, when the boundary speed is the control parameter, the existence of solution is questionable. For example with homogeneous Neumann boundary conditions the existence for the wave equation is an open problem when the variation of the boundary is not monotonic. We derive new results in which the control forces the solution to exist [47] [36] .
Shape optimization theory
Participants : Michel Delfour, Jean-Paul Zolésio.
The ongoing collaboration with the CRM in Montreal (mainly with Professor Michel Delfour) led to several extensions to the theory contained in the book [2] . The emphasis is put on two main aspects: in order to avoid any relaxation approach but to deal with real shape analysis we extend existence results by the introduction of several new families of domains based on fine analysis. Mainly uniform cusp condition, fat conditions and uniform non differentiability of the oriented distance function are studied. Several new compactness results are derived. Also the fine study of Sobolev domains leads to several properties concerning boundaries convergences and boundaries integral convergence under some weak global curvature boundedness [35] .
Control of coupling fluid-structure devices
Participants : John Cagnol, Raja Dziri, Jean-Paul Zolésio.
The use of the transverse vector field governed by the Lie bracket enables us to derive the “first variation” of a free boundary. This result has led to the publication of a book.
An alternate approach to fluid-structure has been developed with P.U.L.V. (J. Cagnol) and the University of Virginia (I. Lasiecka and R. Triggiani, Charlottesville) on stabilization issues for coupled acoustic-shell modeling [63] [28] [55] [30] .
Shape gradient in Maxwell equations
Participants : Pierre Dubois, Jean-Paul Zolésio.
It is well known that in 3D scattering, the geometrical singularities play a special role. The shape gradient in the case of such a singularity lying on a curve in 3D space has been derived mathematically and implemented numerically in the 3D code of France Télécom.
This work with P. Dubois is potentially applicable to more general singularities.
Shape optimization by level set 3D
Participants : Claude Dedeban [ France Télécom ] , Pierre Dubois, Jean-Paul Zolésio.
The inverse scattering problem in electromagnetics is studied
through the identification
or "reconstruction" of the obstacle considered as a smooth surface
in R3 . Through measurement of the
scattered electric field Ed in a zone 
we consider the
classical minimization of a functional 
measuring
the distance beetwen Ed and the actual solution E over .
Then, we introduce the continuous flow mapping Tr , where r is the
disturbance parameter which moves the domain 
in r . We
derive the expression for the shape derivative of the functional, using
a minmax formulation.
Using the Rumsey integral formulation, we solve the Maxwell equation and we compute the shape gradient, verified by finite difference, using the SR3D software (courtesy of the France Telecom company).
Additionally, we have introduced the Level Set representation method in 3 dimensions. This technique, which comes from the image processing community, allows us to construct an optimization method based on the shape gradient knowledge. In this method, the 3D surface, defined by a homogenous triangulation, evolves to reduce the cost functional, easily encompassing certain topological changes. Using this technique, we have studied the inverse problem and evaluated sensibilities w.r.t. quantitative and qualitative criteria.
Shape stabilization of wave equation
Participants : John Cagnol, Jean-Paul Zolésio.
The former results by J.P. Zolésio and C. Trucchi have been extended to more general boundary conditions in order to derive shape stabilization via the energy “cubic shape derivative”. Further extension to elastic shell intrinsic modeling is foreseen.
Passive Shape Stabilization in wave equation. We have developed a numerical code for the simulation of the damping of the wave equation in a moving domain. The cubic shape derivative has been numerically verified through a new approximation taking care of the non autonomeous oprator in the order reduction technique.
Active Shape Morphing. The ongoing collaboration on the stability of wave morphing analysis for drones led to new modeling and sensitivity analyses . Any eigenmode analysis is out of the scope for moving domains as we are faced with time depending operators. Then, we develop a new stability approach directely based ont a "Liapounov decay" by active shape control of the wave morphing. This active control implies a backward adjoint variable and working on the linearized state ( through the transverse vector filed Z which is driven by the Lie brackets) we present a Ricatti-like synthesis for the real time of the morphing.
Array antennas optimization
Participants : Louis Blanchard [ France Telecom R & D ] , Jean-Paul Zolésio.
We are developing a new approach for modeling array antennas optimization. This method integrates a Pareto optimization principle in order to account for the array and side lobes but also the antenna behavior. The shape gradient is used in order to derive optimal positions of the macro elements of the array antenna.
Parametrized level set techniques
Participants : Louis Blanchard, Xavier Hachair, Jean-Paul Zolésio.
Since a 1981 NATO study from the University of Iowa, we know how to define the speed vector field whose flow mapping is used to build the level set of a time-dependent smooth function F(t,x) in any dimension. We consider the Galerkin approach when F(t, .) belongs to a finite-dimensional linear space of smooth functions over the fixed domain D. Choosing an appropriate basis (eigenfunctions, special polynomials, wavelets, ...), we obtain F(t, .) as a finite expansion over the basis with time-dependent coefficients. The Hamilton-Jacobi equation for the shape gradient descent method applied to an arbitrary shape functional (possessing a shape gradient) yields a nonlinear ordinary differential equation in time for these coefficients, which are solved by the Runge-Kutta method of order four. This Galerkin approximation turns out to be very powerful for modeling the topological changes during the domain's evolution. Within the OpRaTel collaboration, L.Blanchard has used extensively the code formely developed by Jérome Picard, also yielding an optimal partionning procedure that is based on the same Galerkin principle but the use of a brute-force calculus to be avoided. Indeed, if the optimal partionning of a domain (e.g. an antenna) consisted in finding a decomposition by 100 subdomains, the level set approach would lead to 100 Hamilton Jacobi equations. We introduced the concept of "multi-saddle" potential function F(t, x) and through the Galerkin technique we follow the evolution of the saddle points. This technique has been succesfully understood thanks to the various testing campaign developed in the OpRaTel collaboration by L. Blanchard and F. Neyme (Thales TAD). The work has permitted to understand the multi-saddle procedure which turns out to require very delicate parameters tuning. We developed a mathematical analysis to justify that the trial-error method and some existence results have been proved for the crossing of the singularity associated with the toplogical change in the Galerkin approximation (here the finite dimensional character is fundamental) [59] .
Shape metrics
Participants : Louis Blanchard, Jean-Paul Zolésio.
We characterize the geodesic for the Courant metric on Shapes. The Courant Metric is described in the book [2] . It furnishes an intrinsic metric for large evolutions. We use the extended weak flow approach in the Euler setting.
It is extended to larger class of sets and using the transverse flow mapping (see the book) we derive evolution equation which characterises the Geodesic for such differentiable metrics [38] [37] .
Applications are being developed for Radar image analysis as well as for various non cylindrical evolution problems including real time control for array antennas.
