Computer software pervades our life but far too much of it contains programming errors (bugs). Software is more and more complex and such errors are unavoidable if programmers are not accompanied with some tools that help auditing software codes. Static analysis is an increasingly popular technique that aims at automatically compute properties of software. These properties then help finding bugs, or proving absence of them. Industrial static analysers are flourishing. Facebook, Google, Microsoft develop their own static analysis tools to help maintaining their huge code base. Critical software industry (aircraft, railways, nuclear, etc.) has embraced the use of advanced static analysis tool as Astrée to companion and sometimes, ligthen their traditional software validation campaigns based on meticulous testing and reviews. Unfortunately, designing advanced static analyses like Astrée requires a very rare expertise in Abstract Interpretation, a foundational landmark in the research area, and implementing these ideas efficiently and correctly is specially tricky. The VESTA project will propose guidance and tool-support to the designers of static analysis, in order to build advanced but reliable static analysis tools. We focus on analyzing low-level softwares written in C, leveraging on the CompCert verified compiler. This compiler toolchain is fully verified in the Coq proof assistant. Verasco is a verified static analyser that I have architected. It analyses C programs and follows many of the advanced abstract interpretation technique developped for Astrée, but it is formally verified. The outcome of the VESTA project will be a platform that help designing other verified advanced abstract interpreters like Verasco, without starting from a white page. We will apply this technique to develop security analyses for C programs. The platform will be open-source and will help the adoption of abstract interpretation techniques.

The aim of this project is to unite efforts of three French teams working on mathematical aspects of turbulence in various physical media. Past successes to tackle turbulence mathematically have been scarce and analytic comprehension has been notoriously difficult. Going further requires new results in Hamiltonian PDE's, probability theory, stochastic PDE, a deep qualitative understanding at a physical level, and possibly insights from numerical simulations. In the last few years, this type of knowledge was used independently by members of this project to obtain complementary original results in turbulence problems. The present joint effort should enable a marked progress in this important field. We will consider mathematical issues related to solutions of the 3D Navier–Stokes equations (the classical turbulence setup) with high Reynolds numbers, and various statistical characteristics of these solutions. We will also consider simpler related models: the two-dimensional stochastic Navier–Stokes equations (2D turbulence, relevant for meteorology and some fields of physics), the stochastic nonlinear Schrödinger equation in dimensions 1, 2, 3 (optical turbulence), the Gross–Pitaevskii equation with stochastic perturbations (turbulence in Bose–Einstein condensation), the stochastic Burgers equation (a popular toy model for the classical turbulence), the Korteweg–de Vries equation with small dissipation and random force (another physical model for turbulence in various media). The project is formed of analytical researches of qualitative properties of solutions for the equations above and for other similar problems. They are supported by numerical studies of the corresponding models. More specifically, our team plans to consider the following questions: 1. Ergodicity for 2D Navier–Stokes equation in a bounded domain with stochastic perturbations localized in the physical or Fourier space 2. The inviscid limit of stationary measures in special cases, such as damped/driven linear or completely integrable equations 3. Qualitative study of dispersive equations with various types of stochastic interventions, such as a random dispersion or a random amplitude of a potential 4. Ergodic behavior of the 2D Euler and Navier–Stokes flows and large-scale structures 5. 2D Navier–Stokes cascades in curved geometry In parallel to the above-mentioned questions, we will investigate the more challenging (and unpredictable) problem of qualitative behavior of solutions for the 3D Navier–Stokes system in bounded and unbounded domains. Only few mathematical results related to the phenomenon of turbulence are known in this context, and there is no good understanding of the problem on the physical level of rigor. Our program will include the investigation of following problems for the 3D Navier–Stokes system and other related and/or simplified equations: 1. Uniform bounds for the local energy for particular classes of solutions 2. Rigorous results on approximation of physically relevant flows by models with a good understanding of the behavior of solutions 3. Investigation of space-time stationary solutions for the Navier–Stokes system with the Ekman damping and other related PDEs 4. Ergodic properties of stochastic models of turbulent transport of inertial particles Further directions for research in the context of the Navier–Stokes system would concern the quantitative study of the direct and inverse cascades and application of the methods of non-equilibrium statistical mechanics to the ergodic theory of nonlinear PDEs The first aim (and main cost) of this four-year project is to extend the existing research effort, by hiring post-docs to work with the involved researchers. The project will also develop long-term relationships between the partners laboratories, each of them internationally recognized in their own field. Finally, the project will fund workshops and meetings to foster international collaborations and discussions.

EpiRL project aims at investigating the combination of epistemic planning and reinforcement learning (RL), by proposing new algorithms that are efficient, adaptive, and capable of computing decisions relying on theory of knowledge and belief. We expect from this approach an efficiency in the generation of epistemic plans, while decisions made RL algorithms will be explainable. Moreover, the algorithms of EpiRL will be tested and evaluated within a real application that exploits autonomous agents. The project will address the weaknesses of both epistemic planning and RL: on the one hand, existing epistemic planning algorithms are costly, do not adapt to the environment, and concepts are hand-crafted and are not learned; on the other hand, in reinforcement learning, agents adapt to their environments but are unable to reason about beliefs of other agents. The newly developed algorithms will combine the strengths of both fields. We propose four workpackages: 1. Study representations of states 2. Develop RL algorithms 3. Study representations of policies 4. Validating our algorithms with our industrial partner DAVI. In particular, we aim at developing a debunking chatbot whose use case will apply to raising awareness about environmental issues.

Manipulating the evolution of atoms and molecules at the quantum level has been a goal from the very beginnings of the laser technology. However, designing laser pulses based on intuition alone did not succeed until the researchers understood that this problem should be attacked with the tools of control theory which greatly contributed to the first positive experimental results. Different applications followed with electromagnetic radiations used for High Harmonic Generation of high frequency lasers, Nuclear Magnetic Resonance applications in medicine, quantum gates design in next generation computers, etc. The field of manipulating molecular evolution is now an emerging technology that requires active input from researchers from different perspectives. In the mathematical formalization the coherent and conservative evolution is determined by the Schrödinger equation that involves the Hamiltonian of the system; in a more general formalism including decoherence and irreversible effects, the state is described by a density matrix operator and evolves according to the Lindblad-Kossakowski master equation. A first and crucial question concerning the manipulation of the quantum evolution by an external field is whether this is possible at all i.e. whether any possible state can be reached with a well-chosen control field; in technical terms this question is called the `controllability of the equation'. One may further refine this question depending on whether the controllability is investigated exactly or approximately, in finite time or asymptotically, in open loop or in closed loop (feedback stabilization)...The study of the control of equations modeling quantum systems is one of the goals of this project. The mere positive conclusion that a control exists does not indicate how to find it in practice. Therefore one needs to formulate numerical algorithms that find the control and are compatible with experimental constraints. Consequently an additional goal of the project is to design such algorithms for specific situations relevant for applications. To be efficient in practice these algorithms have also to take into account the uncertainties and errors and find ways to deal with them. Finally, an estimation (also called `inverse') problem can be formulated: suppose that one does not know the Hamiltonian but can instead measure, for several control fields some aggregate quantity depending on the wave function. How much information on the Hamiltonian can be recovered from such measurements and which are the most efficient procedures to realize this operation? It is well known that these 3 subjects (open loop control, feedback stabilization and estimation) are closely related; which gives additional consistency to our research project. A strong specificity of this proposal is the direct collaboration between physicists, chemists and applied mathematicians. This interaction aims at providing us challenging mathematical problems directly linked to applications. Moreover, the theoretical and numerical advances obtained could be experimentally tested, because the members of the project have direct access to two distinct experimental settings, one at the LKB laboratory (one of the leading physicists at LKB is in the ARMINES-CAS team) and one at Princeton University (group of Prof. H. Rabitz with whom the CEREMADE team has longstanding collaborations). Our goal is to produce mathematical results with an important experimental applicability; which justifies the study of infinite dimensional control systems, numerics, uncertainties, estimation and inversion. Finally, our team presents complementary skills for this purpose: classical and numerical analysis of PDEs, mathematical system theory, quantum physics. Our goal is to achieve the synthesis of these skills.