The goal of this project is to address new directions of research in control theory for partial differential equations, triggered by models from ecology and biology. In particular, our projet will deal with the development of new methods which will be applicable in many applications, from the treatment of cancer cells to the analysis of the thermic efficiency of buildings, and from control issues for the biological control of pests to cardiovascular fluid flows. To achieve these objectives, we will have to solve several theoretical issues in order to design efficient control methods. We have thus targeted four main challenges. The first axis of study concerns the control of parabolic systems, in particular when the control does not act on all the components of the system. In this case, the main difficulty comes from the fact that the components of the system in which the control does not act have to be controlled indirectly through the coupling with the other components of the system. Our goal is to better understand the control properties of such systems, and to develop new robust techniques, which can be applied in particular in nonlinear settings. The second axis will focus on the analysis of the control properties of partial differential equations containing nonlocal terms. Such nonlocal terms appear in many applications, as soon as the dynamics of the system depends on global quantities, or locally averaged quantities. This is for instance the case in population dynamics, where the total number of birth depends on the whole population. New tools should be developed in this context, where the classical techniques developed so far in control theory do not apply. The third point of study aims at addressing in depth major applications involving fluid mechanics. We will in particular study the modeling of the thermic efficiency of buildings and its related control issues, the modeling of cardiovascular flows and stabilization issues around periodic trajectories in models of interactions of viscous or viscoelastic flow (modeling the blood) and an elastic structure (modeling the blood vessels), and the design of waves energy converter of optimal efficiency. The fourth axis of study concerns the control of partial differential equations when the trajectories are required to satisfy some constraints. The study of such question has started very recently and should be developed further and extended to many models, in particular in order to take into account positivity constraints on controlled trajectories, or on some components of it, which are required for the feasibility of the control strategies. This question appears very naturally when some quantities are intrinsically positive, which is the case for instance when they model a population density or a concentration of a chemical reactant. To sum up, with this project, our goal is to reduce the gap between theory and applicative areas and make significant advances for the use of more accurate models and more efficient control designs in real life applications.

Modularity refers to a pattern of connectivity in which elements are grouped into highly connected subsets, modules or building blocks. It is an important property in biology, as it helps a system to "save its current state" while allowing further evolution. Developmental modules are often represented by their physical location and spatial extent in the organism, and they contain informations about the genetic specification of modules. Thus, the properties attributed to modules are: autonomy, discrete organization defined by the expression of specific genes, and occupation of specific physical territories. In vertebrates, the functional and architectural organization of the olfactory system is conserved throughout evolution. The olfactory organ is composed of different types of olfactory sensory neurons, the OSNs, each capable of detecting specific odorant molecules, organized with a specific shape and position, and each expressing a specific set of genes. The hypothesis of the ZOORRO project is that each type of OSNs could be defined by specific modules at key stages of the embryonic development of the olfactory organ. This project aims at breaking down the formation of the olfactory sensory organ into modules: a genetic module and a morphometric-behavioral module. A multidisciplinary consortium of three teams will use a combination of qualitative and quantitative imaging on the zebrafish embryo (Julie Batut, CBI-MCD Toulouse), data science, machine learning and image analysis (Christian Rouvière, CBI Toulouse) together with mathematical models (David Sanchez, INSA Toulouse) to identify the two modules and assemble them to generate a single-cell transcriptomic atlas dynamically linked to cellular behavior and capable of predicting the architecture of a complex biological system: herein, the olfactory sensory organ.

ROMEO is a project which aims to develop a robust system for anomaly detection in drone trajectories based on the use of physics informed neural networks, statistical methods for anomaly detection and quantification of uncertainty through conformal inference methods. In this consortium Thales will bring its expertise in the use of physics informed neural networks and anomaly detection. The "Institut de Mathématiques de Toulouse" (IMT) will bring its expertise in uncertainty evaluation through conformal inference. The massive usage of drones open the path to multiple applications both civil and for defense, including surveillance or smart logistic missions. Such applications may require to use large numbers of drones and in this context, it is crucial to ensure a safe and secure usage of drones through unmanned traffic management (UTM) systems solutions that are both efficient and reliable. In this project, we propose a system which raises alerts for UTM operators. This system raise an alert when an anomaly is detected in the drone trajectory when compared with expected trajectory. The prediction of the normal trajectory will be based on physics informed neural networks, allowing to introduce prior knowledge on the flight dynamics. The anomaly detection will be performed with innovative statistical metrics. The uncertainty on the normal trajectory prediction will be estimated with conformal inference methods. This uncertainty bounds will be integrated in the anomaly detection method in order to provide the operator with trustable alarms.

Following Berger's holonomy classification and Atiyah and Donaldson's achievements in Yang-Mills theory, differential geometers have studied the interactions between variational and algebraic perspectives. Our project combines these traditions in the study of special geometric structures, such as extremal Kähler/Sasaki, special holonomy, generalized geometry and the interplay of all these concepts in Ströminger systems. In practice, these problems belong to gauge theory : a space of connections, a curvature equation to solve, a group of symmetries to control. In each case, the expected outcome is a correspondence between a special geometry and an algebraic condition, as provided by Kobayashi-Hitchin-D-U-Y, which allows to describe the local structure of the moduli space in terms of stability. In addition to constructing new families of examples, our objective is to understand their global topology, deformations and algebraic obstructions.

Anaerobic digestion (AD) is a microbiological process of degradation of the organic matter which produces biogas rich in methane that can be converted into valuable electrical and thermal energy. It is commonly used to manage different types of organic waste at industrial scale using anaerobic digesters. However, this bioprocess is not fully mastered and still has an important potential for improvement. One of the major limitations of AD is the important susceptibility of the microbial communities to changes in operational conditions of the digesters. It can lead to unstable methane formation. Controlling AD microbial community stability, though, is not a trivial task. Knowledge on the determinants of anaerobic microbial process stability (i.e. the conditions and the succession of microbial events that allow maintaining a balance after a disruption or, on the contrary, that generate a domino effect leading to total failure) over time is still missing. Emerging omics high-throughput approaches can now lead to unprecedented data to portray AD microbiome. Metagenomics, metatranscriptomics, metaproteomics and metabolomics enable to describe a community at different levels (genes, gene expression, and metabolites production). Appropriate and efficient analytical methods are required to analyse these big and complex data and unravel the intricate networks of functional processes of AD. Novel computational and statistical methods are progressively becoming available to fully harvest and integrate these complex datasets. In this context, the aim of STABILICS is to conduct the first sets of high-throughput multi-omics longitudinal experiments, with an unprecedented sampling depth, in anaerobic digesters under constant environmental parameters or subject to different model perturbations created by the addition of NaCl. Experiments in lab-scale semi-continuous reactors will be set-up and monitored in the long run (more than one year). Two levels of analysis will be applied. 1) A high frequency monitoring of different descriptors of microbiota activity, where non-targeted metabolomics and isotopic analyses will characterise the degradation pathways and metabarcoding of RNA and DNA will target both active and present microorganisms. 2) An in-depth monitoring of microbiota functioning with both metagenomics and metatranscriptomics on selected samples and conditions. These unprecedented sets of data will be thoroughly analysed and integrated using cutting-edge statistical methods. For example, multivariate dimension reduction methods will be used for data mining, omics integration and feature selection; specific analytical framework for longitudinal data will be developed. The objectives of this interdisciplinary project will be 1) to evaluate at different omics levels the dynamics of AD microbiome in long term and replicated time course experiments, 2) to describe the succession of events that, under stress, leads to microbiota equilibrium unbalance and digester disruption or on the contrary microbiota equilibrium preservation and maintenance of stability, 3) to propose an original analytical framework of multi-omics longitudinal studies accounting for temporality, and 4) to deliver generic knowledge to understand the determinants of perturbations.