Emmanuel Trélat - Selected Publications#

J.A. Carrillo, D. Kalise, F. Rossi, and E. Trélat, Controlling swarms towards flocks and mills, SIAM J. Control Optim. 60 (2022), no. 3, 1863–1891.

For several years, E. Trélat has been actively collaborating with various colleagues (L. Boudin, M. Caponigro, J. Carrillo, M. Fornasier, D. Kalise, B. Piccoli, F. Rossi, F. Salvarani, see in particular the above most recent article) on questions of collective dynamics, self-organization and control: movements of birds and fish, evolution of opinions, convergence towards a consensus in a graph

His first contribution in a series of articles was with M. Caponigro, M. Fornasier and B. Piccoli, and was published in Math. Models Methods Appl. Sci. in 2015.

In these papers, E. Trélat has introduced the notion of sparse control, modelling the fact that it is only possible to act on such systems (comprising a large number of agents) through very localised actions. He obtained the first controllability results for these systems, first in finite dimension (Cucker-Smale systems) and then in infinite dimension (kinetic equations), by mean field limit when the number of agents tends to infinity.

In the above-mentioned recent paper, on a swarming model involving self-propulsion and potentials, he shows for instance how to steer a group of agents, like animals or opinions, from a flock to a mill configuration, i.e., a periodic motion (this is the first time in the literature), by designing explicit feedback sparse control laws using fine dynamical features along heteroclinic connections.

Y. Colin de Verdière, L. Hillairet, and E. Trélat, Spectral asymptotics for sub-Riemannian Laplacians, I: Quantum ergodicity and quantum limits in the 3-dimensional contact case, Duke Math. J. 167 (2018), no. 1, 109–174.

and Y. Colin de Verdière, L. Hillairet, and E. Trélat, Spectral asymptotics for sub-Riemannian Laplacians, Preprint, arXiv:2212.02920, https://arxiv.org/abs/2212.02920

Having discovered quantum chaos theory in his work on optimal observability (see below), in recent years E. Trélat, in collaboration with Y. Colin de Verdière and L. Hillairet, has immersed himself in the spectral analysis of sub-elliptic Laplacians, opening up a new path in sub-Riemannian in connection with microlocal analysis. They were the first to obtain a quantum ergodicity theorem in the sub-elliptic case (the article published in Duke Math. J.) and then, in a very recent major work (the preprint), to Weyl laws in a very general sub-Riemannian framework. It opens numerous new perspectives and new fields of applications.

S. Arguillère, E. Trélat, A. Trouvé, and L. Younes, Shape deformation analysis from the optimal control viewpoint, J. Math. Pures Appl. 104 (2015), no. 1, 139–178.

In this article, the authors model the problems of shape analysis in the general framework of optimal control, thus making it possible to consider constraints on the movements of these shapes. To this end, they developed aspects of optimal control in infinite dimension, acting on the flow of diffeomorphisms whose controls are the (time-varying) vector fields that generate them. There are numerous applications in particular in the medical field (heart movements, for example). This work marks a turning point in the theory, since until now only Riemann geometry has been used in this context. To model non-holonomic constraints, they use sub-Riemannian geometry (in this case, in infinite dimension, which is also new), and more generally optimal control.

C. Pouchol, J. Clairambault, A. Lorz, and E. Trélat, Asymptotic analysis and optimal control of an integro- differential system modelling healthy and cancer cells exposed to chemotherapy, J. Math. Pures Appl. 116 (2018), 268–308.

E. Trélat carries out an optimal control study aimed at improving drug treatments for cancer. He uses two coupled integro-differential equations model reflecting the temporal evolution of healthy and cancerous cells resisting to the treatment according to a phenotypic variable. His study confirms the observation by medical experts that the periodic treatment strategy can be greatly improved and gives an explicit (quasi-)optimal treatment. A remarkable result which is of course of great importance.

Y. Privat, E. Trélat, and E. Zuazua, Optimal shape and location of sensors for parabolic equations with random initial data, Arch. Ration. Mech. Anal. 216 (2015), no. 3, 921–981.

and

Y. Privat, E. Trélat, and E. Zuazua, Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 5, 1043–1111.

In collaboration with Y. Privat and E. Zuazua, E. Trélat studies in these two articles the problem of optimising the observation, control or stabilisation domain for PDEs.Tthis involves optimising not only the placement of sensors/actuators but also their shape (optimal design). They adopt a probabilistic approach by modifying the usual notion of observability inequality, which is too pessimistic because it is deterministic (representing the worst case). In practice, we want things to go as well as possible, 'on average', over a large number of experiments. For parabolic PDEs, they demonstrate in the first article the existence and uniqueness of the optimal domain and obtain a spectral characterisation by a finite number of modes, thus giving, in addition, a method of calculation that is both simple and efficient. For hyperbolic PDEs, of the wave or Schrödinger type, the situation is more complicated : all the modes count, and their second article reveals a close and unexpected links with the asymptotic properties of the eigenfunctions (quantum ergodicity) and quantum chaos theory. These two articles are master pieces.

E. Trélat and E. Zuazua, The turnpike property in finite-dimensional nonlinear optimal control, J. Differential Equations 258 (2015), no. 1, 81–114.

One of the striking properties for long-time optimal control problem is the turnpike property, on which E. Trélat has contributed a great deal, notably with this article with E. Zuazua. The turnpike principle, discovered by Samuelson (Nobel Prize winner in economics), states that, in a long-time optimal control problem, the optimal trajectory remains "essentially static", this stationary state being itself the solution to a static optimisation problem. This is a phenomenon often encountered in the life sciences. In this article E. Trélat proves the validity of this principle under very general hypotheses, in finite or infinite dimension. E. Trélat thus demonstrates the universality of this phenomenon and its great practical importance in the numerical implementation of many problems in optimal control and dynamic shape optimisation.

E. Trélat, Optimal control and applications to aerospace : some results and challenges, J. Optim. Theory Appl. 154 (2012), no. 3, 713–758.

Emmanuel Trélat is also an expert of numerical schemes for optimal control problems and has brought important contributions on genuine industrial applications, in particular in aerospace as shown in this article. What is probably his most striking result in industrial collaborations is his work with EADS Astrium on the development of a software for real-time trajectory optimization of Ariane V launchers. Instead of using standard optimization approaches, he implemented a new method based on a combination of tools of optimal control theory, a refined geometric insight of the geodesics, continuation methods and other numerical tools. This is an important breakthrough because it is usually well known that it is very difficult to implement efficiently indirect methods based on the Pontryagin maximum principle. He carries out an outstanding solution to the difficult problem of getting an automatic, reliable and (almost) instantaneous tool in order to compute trajectories of launchers for the problem of minimal consumption, thus giving an important advance to EADS in this competitive field.

Y. Chitour, F. Jean, and E. Trélat, Genericity results for singular curves, J. Differential Geom. 73 (2006), no. 1, 45–73.

In this paper E. Trélat proves, in collaboration with Y. Chitour and F. Jean, genericity results on the absence of minimizing singular trajectories (not reduced to a point) in optimal control. Singular trajectories are trajectories whose the linearised control around them is not controllable. It has important theoretical and numerical consequences. From a theoretical point of view, the absence of minimising singularities guarantees key properties of regularity of the value function and viscosity solutions of certain classes of Hamilton-Jacobi equations, making it possible to establish that their singular locus is a stratified subvariety of codimension 1. Numerically, this allows to guarantee the convergence of numerical algorithms such as the shooting method or homotopy methods, which is of course very important.