The fruitful application of logical methods in several areas of computer science, epistemology and artificial intelligence has resulted in an explosion of new logics. These logics are more expressive than classical logic, allowing finer distinctions and a direct representation of notions that do not find a natural place in classical logic. Logics are used to express different modes of truth (modal logics) and other types of reasoning including hypothetical and plausible reasoning (conditional logics), reasoning about knowledge (epistemic logics), and separation and sharing of resources (bunched implications logics). In addition to formalizing reasoning in this way, logics are also used to model various systems and to prove properties about them leading to applications in checking correctness and safe behaviour. In this project we consider those logics that are variants and generalizations of modal logics (inclusive of all the logics listed above) and characterized by variants of Kripke semantics; they find applications specifically in the areas of formal verification, epistemology and knowledge representation. Our investigation will focus on the proof-theory of these logics. Proof-theory provides a constructive approach to investigating fundamental meta-logical and computational properties of a logic through the design and the study of calculi (formal proof systems) with suitable properties (analyticity). Analytic calculi are also the base for developing practical reasoning tools such as theorem provers and proof assistants. In the literature of the last 30 years, several formal proof systems, generalizing the original sequent calculi by Gentzen, have been proposed to provide analytic calculi for modal and related logics; among them hypersequent calculi, labelled calculi and display calculi. The proof systems we study here fall into two categories: internal calculi, in which every basic object of the calculus can be read as a formula of the language of the logic, and external calculi where the basic objects are formulas of a more expressive language which partially encode the semantics (meaning) of the logic. The success of this investigation is varied: for some important classes of logics, no internal calculi are known, for others no terminating or optimal external calculi are known. The internal and external calculi reflect the two different ways of presenting a logic: syntactically and semantically. Both presentations are useful: they exhibit distinct properties and reveal different facets of the logic. The relationships between internal and external calculi are largely unexplored and their investigation is our main objective. We intend to systematically study the relationships between internal and external calculi with the aim of transferring the advantages from one type of calculi into the other. We think that such a study will shed light on the relationship between provability in syntactic and semantic-based calculi, enable the transfer of proof-theoretic properties between different calculi and lead to the discovery of internal calculi for logics that do not yet enjoy them. These new internal calculi will be helpful for the solution of several important theoretical problems including interpolation and conservativity. Indeed, the question of decidability is also still open for many logics and a main obstacle is the lack of an analytic internal calculus. The TICAMORE project will clarify the relationship between the two fundamental and historically distinct approaches, thus promoting the unification and cross-fertilization of new ideas between practitioners in the two communities, leading to new insights into the proof-theory of modal and related logics. Finally the project will contribute to the development of new automated reasoning tools to be applied in knowledge representation and in the formal verification of system properties.

In recent years, the study of collective behavior of a crowd of autonomous agents has drawn a great interest from scientific communities, e.g. in civil engineering (for emergency egress and traffic problems), robotics (coordination of robots), computer science and sociology (social networks), and biology (crowds of animals). In particular, it is well-known that some simple rules of interaction between agents can provide the formation of special patterns, like in formations of bird flocks, lines, etc... This phenomenon is often referred to as self-organization or emerging behavior. Beside the problem of analyzing the collective behavior of such systems, it is now interesting to understand what changes of behavior can be induced by an external agent (e.g. a policy maker) to the crowd. For example, one can try to enforce the creation of patterns when they are not formed naturally, or break the formation of such patterns. This is the problem of control of crowds. Two control methods are studied in the project in different frameworks. The first (TASK 1) is the control of crowds with leaders, i.e. by acting on a small and fixed number of agents. The resulting control has the clear advantage of focusing on few agents, then naturally producing sparse (or parsimonious) strategies. The second method (TASK 2) is the control with local policies, modeled as by perturbation of the crowd dynamics in a small subset of the configuration space, modeling external interventions. This model also produces sparse strategies, in terms of the size of the control domain. The key application of control of traffic models will be addressed in TASK 3. The goal is both to solve challenging specific problems of traffic control, and to use such application as a benchmark for the theory developed in TASKS 1-2. Indeed, we will study problems of control of traffic both with autonomous vehicles acting as leaders and with time-varying speed limits acting as localized policies. Problems of control of crowds addressed in the project can be stated in different mathematical frameworks, depending on the model used to describe the crowd dynamics. We address it in three different mathematical frameworks: microscopic models (sizable finite-dimensional dynamical systems), macroscopic models (transport partial differential equations), and multi-scale models (measure evolutions). The control problems will be effectively studied by building bridges between the three mathematical frameworks, by studying in which cases control strategies can be translated from one setting to another. Several methods for such approach were developed and improved by the project team. In particular: - mean-field limits, that allow to pass from microscopic to macroscopic models. In the case of controlled systems, such limit must be handled with care, and the P.I. and collaborators have developed specific successful techniques. - multi-scale models, that mix microscopic and macroscopic approaches. The team solved optimal control problems in the case of leaders-and-followers model, that can be adapted to the measure setting. - the Wasserstein distance and its generalizations, unifying the three frameworks. - numerical schemes developed by the P.I. and collaborators for the three frameworks. The researches developed in this project will be carried out by a team based in Marseille. The project aims to establish a small strong group focused on control of multi-agent systems, connecting approaches from control and applied mathematics. The strength of our team is its interdisciplinarity, that permits to merge different techniques (geometric control, control of partial differential equations, systems theory) and a variety of tools to give a significant contribution in solving challenging problems of crowd control.

For the five main identified risk domains, the SPID project aims at leading a technical-operational threat analysis, taking into account each sensitive site typology, each potential site to protect, various detection and neutralizations possible scenarios and finally analyze vulnerability of the proposed system. The project includes producing a demonstration system operating multiple types of sensors, with innovating sensors: acoustics, optronics, radio-goniometry. Sensors are fitted with built-in processing capability, connected in a network, they will broadcast data towards a central server designed to manage and store all communication data. The server will be able to assure data processing and fusion, to display toward the operator via MMI, real-time vision of the situation and access to all evidence data. A link will allow to forward all these data towards a command and control system, fixed or mobile governmental type or, towards a control room of the site to protect. The system allows surveillance and detection, target recognition and classification, and if possible, identification, geolocalization and tracking in order to neutralize the threat. The system will be operated and evaluated on capacitive and operational criteria according threat scenarios defined during the operational analysis, with real UAVs flights of fixed or rotary wings types. During these tests, operational capacity provided by radars will be also tested and a flight campaign will be arranged in order to invite industry to test their solutions in optronics, radar and other technologies. These tests will be defined in collaboration with a panel of users representing authorities involved in fighting against these threats as well as operators of industrial sensitive sites or vast sites receiving public (entertainment), in order to cover all kinds of situations. For the neutralization aspect, a state of the art study on technical and operational will be conducted for all possible solutions, with advantages and drawbacks of each solution. Radio-goniometry will be included in the tests to check its contribution in terms of neutralization of standard UAVs. Juridical and regulation aspects including sanitary, respect of individual liberty and private life, sociological and organizational aspects towards public community will be addressed by a study. SPID project will thus allow to define a first approach of an operational system providing all weather detection capability. Neutralization capability of non standard UAVs will be addressed later exploiting the best solutions deriving from the state of the art study. The goal consists in marketing an efficient alert system for an affordable price for the authorities and operators of numerous sensitive sites present in France, in a first step, and quickly after to market it internationally. This project represents for the SMEs members of SPID, as a platform to support their innovation to expand on new markets thanks to their excellent operational knowledge provided with the relation with the users who will validate the proposed solutions. The market study and the business strategy including scientific communication tools but also events dedicated to security professionals will allow to maximize the economic impact for the SMEs and obtain a direct benefit for employment increase. SPID members benefit already from a recognized knowledge and significant experience in security domain, the wide panel of end-users representing recognized units and SPID team demonstrate high technical value and a perfect match between partners.

Sub-Riemannian manifolds provide an important mathematical model for many problems involving some nonholonomic constraints. Since several years, there has been an impressive revival of interest in sub-Riemannian geometry (in short, SR geometry), together with many emerging interactions. SR geometry is of course interesting in itself and raises a number of fascinating mathematical challenges, such as the study of SR heat kernels, hypoelliptic diffusions, volumes and singularities. But SR geometry has also a wide potential of interactions. A classical (we could almost say, historical) field of applications of SR geometry is robotics and motion planning, in relationship with geometric control theory, but in the recent years SR geometry has appeared relevant in new fields, with striking issues, such as in optimal transport, in image reconstruction and geometry of vision, or even more recently, in quantum physics and in shape analysis. The study of heat diffusion in Riemannian geometry has been strongly developed, both from the probabilistic point of view (Brownian motions, Wiener measures), and for the study of direct and inverse spectral problems (Kac's famous ``can one hear the shape of a drum?'', Weyl asymptotic formula). A similar study in the SR case (i.e., for SR Laplacians) is still rather incomplete and important for applications. Generalizing the definition of Laplace-Beltrami operator in Riemannian geometry, SR Laplacians are defined as the divergence (for some measure) of the horizontal gradient (for some SR metric). In this perspective, a domain which is of interest to several mathematical communities concerns the study of the small-time SR heat kernel asymptotics, with many interesting issues that we want to investigate: local Weyl law and quantum ergodicity within the SR context, study of Weyl measures, propagation of singularities for SR wave equations and impact of abnormal geodesics, inverse spectral problems and identification of new spectral invariants. This is the content of Task 1 of our project. SR Laplacians depend on the choice of the measure, and in SR geometry there are several intrinsic choices, such as Hausdorff or Popp volumes. The regularity of those volumes, one with respect to each other, raises surprisingly difficult questions, in particular at singularities of the horizontal distribution of the SR structure. Such singularities cause barrier phenomena for SR heat and Schrodinger flows. The (new) notion of SR curvature seems to be related to spectral invariants that we want to identify. The case of Carnot groups is of a particular interest. We want also to investigate the concept of horizontal holonomy, which is the SR version of the holonomy group associated with a connection. This is the content of Task 2. Task 3 gathers interactions of SR geometry with other fields. The first consists of transportation problems involving nonholonomic constraints. The problem of existence and uniqueness of an optimal transport map on a general complete SR manifold is open. It is related to curvature phenomena in SR manifolds and to isoperimetric problems in these spaces. The second is that SR geometry (and in particular, hypoelliptic diffusion) provides a relevant framework to geometry of vision, as it has been recently discovered, due to the fact that the architecture of connections in the visual cortex seems to reflect a contact distribution. Finally, even more recently SR geometry has emerged as well in shape analysis, where pattern matching is achieved in the group of so-called horizontal diffeomorphisms, thus modelling image analysis problems in which motions are submitted to nonholonomic constraints.