Francesco Rossi

• Ph.D. in Applied Mathematics
• Professor at Università Iuav di Venezia

We consider multi-agent systems with cooperative interactions and study the convergence to consensus in the case of time-dependent lack of interaction. We prove a new condition ensuring consensus: we define a graph in which directed arrows correspond to connection functions that converge (in the weak sense) to some function with a positive integral...

For networked systems, Persistent Excitation and Integral Scrambling Condition are conditions ensuring that communication failures between agents can occur, but a minimal level of service is ensured. We consider cooperative multi-agent systems satisfying either of such conditions. For first-order systems, we prove that consensus is attained. For se...

Let a finite set of interacting particles be given, together with a symmetry Lie group $G$. Here we describe all possible dynamics that are jointly equivariant with respect to the action of $G$. This is relevant e.g., when one aims to describe collective dynamics that are independent of any coordinate change or external influence. We particularize...

We show the existence of Lipschitz-in-space optimal controls for a class of mean-field control problems with dynamics given by a non-local continuity equation. The proof relies on a vanishing viscosity method: we prove the convergence of the same problem where a diffusion term is added, with a small viscosity parameter. By using stochastic optimal...

Numerous research projects have faced the problem of the interpretation of post-disaster reconstructions. Several contributions have approached the problem in terms of identifying urban-setting reconstruction models, some attempting a systemization on a historiographic basis. To date, however, there has been no comprehensive work aimed at developin...

Models of social influence may present discontinuous dynamical rules, which are unavoidable with topological interactions, i.e. when the dynamics is the outcome of interactions with a limited number of nearest neighbors. Here, we show that classical solutions are not sufficient to describe the resulting dynamics. We first describe the time evolutio...

We study the possibility of defining a distance on the whole space of measures, with the property that the distance between two measures having the same mass is the Wasserstein distance, up to a scaling factor. We prove that, under very weak and natural conditions, if the base space is unbounded, then the scaling factor must be constant, independen...

Self-organization and control around flocks and mills is studied for second-order swarming systems involving self-propulsion and potential terms. It is shown that through the action of constrained control, it is possible to control any initial configuration to a flock or a mill. The proof builds on an appropriate combination of several arguments: t...

The problem of identifying models of post-disaster reconstruction is an issue that has been dealt with in depth in a number of works, mostly dedicated to individual cases or to the comparison of models. There are also a number of works that have attempted a systematisation on an historiographic basis. To date, however, there is no overall work aime...

We study the possibility of defining a distance on the whole space of measures, with the property that the distance between two measures having the same mass is the Wasserstein distance, up to a scaling factor. We prove that, under very weak and natural conditions, if the base space is unbounded, then the scaling factor must be constant, independen...

The evoluted set is the set of configurations reached from an initial set via a fixed flow for all times in a fixed interval. We find conditions on the initial set and on the flow ensuring that the evoluted set has negligible boundary (i.e. its Lebesgue measure is zero). We also provide several counterexample showing that the hypotheses of our theo...

We show the existence of Lipschitz-in-space optimal controls for a class of mean-field control problems with dynamics given by a non-local continuity equation. The proof relies on a vanishing viscosity method: we prove the convergence of the same problem where a diffusion term is added, with a small viscosity parameter. By using stochastic optimal...

Social dynamics models may present discontinuities in the right-hand side of the dynamics for multiple reasons, including topology changes and quantization. Several concepts of generalized solutions for discontinuous equations are available in the literature and are useful to analyze these models. In this chapter, we study Caratheodory and Krasovsk...

The evoluted set is the set of configurations reached from an initial set via a fixed flow for all times in a fixed interval. We find conditions on the initial set and on the flow ensuring that the evoluted set has negligible boundary (i.e. its Lebesgue measure is zero). We also provide several counterexample showing that the hypotheses of our theo...

Bounded-confidence models in social dynamics describe multi-agent systems, where each individual interacts only locally with others. Several models are written as systems of ordinary differential equations (ODEs) with discontinuous right-hand side: this is a direct consequence of restricting interactions to a bounded region with non-vanishing stren...

Social dynamics models may present discontinuities in the right-hand side of the dynamics for multiple reasons, including topology changes and quantization. Several concepts of generalized solutions for discontinuous equations are available in the literature and are useful to analyze these models. In this chapter, we study Caratheodory and Krasovsk...

We study a family of optimal control problems in which one aims at minimizing a cost that mixes a quadratic control penalization and the variance of the system, both for finite sets of agents and for the limit problem as their number goes to infinity. While the solution for finitely many agents always exists in a unique and explicit form, the behav...

We consider a mean-field control problem with linear dynamics and quadratic control. We apply the vanishing viscosity method: we add a (regularizing) heat diffusion with a small viscosity coefficient and let such coefficient go to zero. The main result is that, in this case, the limit optimal control is exactly the optimal control of the original p...

Self-organization and control around flocks and mills is studied for second-order swarming systems involving self-propulsion and potential terms. It is shown that through the action of constrained control, is it possible to control any initial configuration to a flock or a mill. The proof builds on an appropriate combination of several arguments: L...

Self-organization and control around flocks and mills is studied for second-order swarming systems involving self-propulsion and potential terms. It is shown that through the action of constrained control, is it possible to control any initial configuration to a flock or a mill. The proof builds on an appropriate combination of several arguments: L...

We study a family of optimal control problems in which one aims at minimizing a cost that mixes a quadratic control penalization and the variance of the system, both for finitely many agents and for the mean-field dynamics as their number goes to infinity. While solutions of the discrete problem always exist in a unique and explicit form, the behav...

Bounded-confidence models in social dynamics describe multi-agent systems, where each individual interacts only locally with others. Several models are written as systems of ordinary differential equations with discontinuous right-hand side: this is a direct consequence of restricting interactions to a bounded region with non-vanishing strength at...

We prove the existence of a universal gap for minimum time controllability of finite dimensional quantum systems, except for some basic representations of spin groups. This is equivalent to the existence of a gap in the diameter of orbit spaces of the corresponding compact connected Lie group unitary actions on the Hermitian spheres.

In this work, we study the minimal time to steer a given crowd to a desired configuration. The control is a vector field, representing a perturbation of the crowd velocity, localized on a fixed control set. We will assume that there is no interaction between the agents. We give a characterization of the minimal time both for microscopic and macrosc...

We introduce the optimal transportation interpretation of the Kantorovich norm on thespace of signed Radon measures with finite mass, based on a generalized Wasserstein distancefor measures with different masses.With the formulation and the new topological properties we obtain for this norm, we proveexistence and uniqueness for solutions to non-loc...

In this paper, we provide a sufficient condition for the Lipschitz-in-space regularity for solutions of optimal control problems formulated on continuity equations. Our approach involves a novel combination of mean-field approximation results for infinite-dimensional multi-agent optimal control problems along with an existence result of locally opt...

In this paper, we extend and complete the classification of the generic singularities of the 3D-contact sub-Riemmanian conjugate locus in a neighborhood of the origin.

In this paper, we study sufficient conditions for the emergence of asymptotic consensus and flocking in a certain class of non-linear generalised Cucker-Smale systems subject to multiplicative communication failures. Our approach is based on the combination of strict Lyapunov design together with the formulation of a suitable persistence condition...

We provide a mean-field description for a leader–follower dynamics with mass transfer among the two populations. This model allows the transition from followers to leaders and vice versa, with scalar-valued transition rates depending nonlinearly on the global state of the system at each time. We first prove the existence and uniqueness of solutions...

We provide a mean-field description for a leader-follower dynamics with mass transfer among the two populations. This model allows the transition from followers to leaders and vice versa, with scalar-valued transition rates depending nonlinearly on the global state of the system at each time. We first prove the existence and uniqueness of solutions...

We provide a mean-field description for a leader-follower dynamics with mass transfer among the two populations. This model allows the transition from followers to leaders and vice versa, with scalar-valued transition rates depending nonlinearly on the global state of the system at each time. We first prove the existence and uniqueness of solutions...

In this paper, we prove the existence of a universal gap for controllability (minimum time) of finite dimensional quantum systems. This is equivalent to the existence of a gap in the diameter of orbit spaces of compact connected Lie group unitary actions on the Hermitian spheres. It is known that such a gap does not exist for finite group orthogona...

We prove a Pontryagin Maximum Principle for optimal control problems in the space of probability measures, where the dynamics is given by a transport equation with non-local velocity. We formulate this first-order optimality condition using the formalism of subdifferential calculus in Wasserstein spaces. We show that the geometric approach based on...

In this paper, we extend and complete the classification of the generic singularities of the 3D-contact sub-Riemmanian conjugate locus in a neighbourhood of the origin.

In this work, we study the minimal time to steer a given crowd to a desired configuration. The control is a vector field, representing a perturbation of the crowd velocity, localized on a fixed control set. We give a characterization of the minimal time both for microscopic and macroscopic descriptions of a crowd. We show that the minimal time to s...

We introduce a new formulation for differential equation describing dynamics of measures on an Euclidean space, that we call Measure Differential Equations with sources. They mix two different phenomena: on one side, a transport-type term, in which a vector field is replaced by a Probability Vector Field, that is a probability distribution on the t...

This chapter revises some modeling, analysis, and simulation contributions for crowd dynamics using time-evolving measures. Two key features are strictly related to the use of measures: on one side, this setting permits to generalize both microscopic and macroscopic crowd models. On the other side, it allows an easy description of multi-scale crowd...

For control-affine systems with a proper Lyapunov function, the classical Jurdjevic–Quinn procedure (see Jurdjevic and Quinn, 1978) gives a well-known and widely used method for the design of feedback controls that asymptotically stabilize the system to some invariant set. In this procedure, all controls are in general required to be activated, i.e...

We study the controllability of a Partial Differential Equation of transport type, that arises in crowd models. We are interested in controlling it with a control being a vector field, representing a perturbation of the velocity, localized on a fixed control set. We prove that, for each initial and final configuration, one can steer approximately o...

We study the controllability of a Partial Differential Equation of transport type, that arises in crowd models. We are interested in controlling it with a control being a vector field, representing a perturbation of the velocity, localized on a fixed control set. We prove that, for each initial and final configuration, one can steer approximately o...

We derive a Maximum Principle for optimal control problems with constraints given by the coupling of a system of ODEs and a PDE of Vlasov-type. Such problems arise naturally as Γ-limits of optimal control problems subject to ODE constraints, modeling, for instance, external interventions on crowd dynamics. We obtain these first-order optimality con...

We study controllability of a Partial Differential Equation of transport type, that arises in crowd models. We are interested in controlling such system with a control being a Lipschitz vector field on a fixed control set omega. We prove that, for each initial and final configuration, one can steer one to another with such class of controls only if...

We present a new chamber matching algorithm, which is completely data-driven and unsupervised, and designed for the semiconductor industry. The behavior of an equipment is classified as different when the shape of the time series given by one of the sensors is significantly different. Shape comparison is performed using linear regression, that auth...

In the present chapter, we study the emergence of global patterns in large groups in first- and second-order multiagent systems, focusing on two ingredients that influence the dynamics: the interaction network and the state space. The state space determines the types of equilibrium that can be reached by the system. Meanwhile, convergence to specif...

Cooperative systems are systems in which the forces among agents are non-repulsive. The free evolution of such systems can tend to the formation of patterns, such as consensus or clustering, depending on the properties and intensity of the interaction forces between agents. The kinetic cooperative systems are obtained as the mean field limits of th...

In this work, we study the minimal time to steer a given crowd to a desired configuration. The control is a vector field, representing a perturbation of the crowd velocity, localized on a fixed control set. We characterize the minimal time for a discrete crowd model, both for exact and approximate controllability. This leads to an algorithm that co...

In this work, we study the minimal time to steer a given crowd to a desired configuration. The control is a vector field, representing a perturbation of the crowd velocity, localized on a fixed control set. We characterize the minimal time for a discrete crowd model, both for exact and approximate controllability. This leads to an algorithm that co...

We study controllability of a Partial Differential Equation of transport type, that arises in crowd models. We are interested in controlling such system with a control being a Lipschitz vector field on a fixed control set $\omega$. We prove that, for each initial and final configuration, one can steer one to another with such class of controls only...

We consider nonlinear transport equations with non-local velocity describing the time-evolution of a measure. Such equations often appear when considering the mean-field limit of finite-dimensional systems modeling collective dynamics. We address the problem of controlling these equations by means of a time-varying bounded control action localized...

We consider nonlinear transport equations with non-local velocity, describing the time-evolution of a measure, which in practice may represent the density of a crowd. Such equations often appear by taking the mean-field limit of finite-dimensional systems modelling collective dynamics. We first give a sense to dissipativity of these mean-field equa...

The Wasserstein distances W p (p \({\geqq}\) 1), defined in terms of a solution to the Monge–Kantorovich problem, are known to be a useful tool to investigate transport equations. In particular, the Benamou–Brenier formula characterizes the square of the Wasserstein distance W 2 as the infimum of the kinetic energy, or action functional, of all vec...

Among the main actors of organism development there are morphogens, which are signaling molecules diffusing in the developing organism and acting on cells to produce local responses. Growth is thus determined by the distribution of such signal. Meanwhile, the diffusion of the signal is itself affected by the changes in shape and size of the organis...

We prove the existence and pointwise lower and upper bounds for the fundamental solution of the degenerate second order partial differential equation related to Geman-Yor stochastic processes, that arise in models for option pricing theory in finance. Lower bounds are obtained by using repeatedly an invariant Harnack inequality and by solving an as...

We prove the existence and pointwise lower and upper bounds for the fundamental solution of the degenerate second order partial differential equation related to Geman-Yor stochastic processes, that arise in models for option pricing theory in finance. Lower bounds are obtained by using repeatedly an invariant Harnack inequality and by solving an as...

In the present chapter we study the emergence of global patterns in large groups in first and second-order multi-agent systems, focusing on two ingredients that influence the dynamics: the interaction network and the state space. The state space determines the types of equilibrium that can be reached by the system. Meanwhile, convergence to specifi...

Among the main actors of organism development there are morphogens, which are signaling molecules diffusing in the developing organism and acting on cells to produce local responses. Growth is thus determined by the distribution of such signal. Meanwhile, the diffusion of the signal is itself affected by the changes in shape and size of the organis...

We study an optimal control problem for traffic regulation via variable speed limit. The traffic flow dynamics is described with the Lighthill-Whitham-Richards (LWR) model with Newell-Daganzo flux function. We aim at minimizing the $L^2$ quadratic error to a desired outflow, given an inflow on a single road. We first provide existence of a minimize...

We study an optimal control problem for traffic regulation via variable speed limit. The traffic flow dynamics is described with the Lighthill-Whitham-Richards (LWR) model with Newell-Daganzo flux function. We aim at minimizing the $L^2$ quadratic error to a desired outflow, given an inflow on a single road. We first provide existence of a minimize...

We prove existence and uniqueness of solutions to a transport equation modelling vehicular traffic in which the velocity field depends non-locally on the downstream traffic density via a discontinuous anisotropic kernel. The result is obtained recasting the problem in the space of probability measures equipped with the $\infty$-Wasserstein distance...

In this paper, we introduce the concept of Developmental Partial Differential Equation (DPDE), which consists of a Partial Differential Equation (PDE) on a time-varying manifold with complete coupling between the PDE and the manifold's evolution. In other words, the manifold's evolution depends on the solution to the PDE, and vice versa the differe...

The well-known Cucker-Smale model is a microscopic system reproducing the alignment of velocities in a group of autonomous agents. Here, we focus on its mean-field limit, which we call the continuous Cucker-Smale model. It is a transport partial differential equation with nonlocal terms. For some choices of the parameters in the Cucker-Smale model...

We derive a Maximum Principle for optimal control problems with constraints given by the coupling of a system of ODEs and a PDE of Vlasov-type. Such problems arise naturally as ${\Gamma}$-limits of optimal control problems subject to ODE constraints, modeling, for instance, external interventions on crowd dynamics. We obtain these first-order optim...

In this paper we deal with a social dynamics model, where one controls a small number of leaders in order to influence the behavior of the whole group (leaders and followers). We first provide a general mathematical framework to deal with optimal control of the microscopic problem, where the number of agents is finite, and its mean-field limit with...

The well-known Cucker-Smale model is a macroscopic system reflecting flocking, i.e. the alignment of velocities in a group of autonomous agents having mutual interactions. In the present paper, we consider the mean-field limit of that model, called the kinetic Cucker-Smale model, which is a transport partial differential equation involving nonlocal...

We introduce the rigorous limit process connecting finite dimensional sparse optimal control problems with ODE constraints, modeling parsimonious interventions on the dynamics of a moving population divided into leaders and followers, to an infinite dimensional optimal control problem with a constraint given by a system of ODE for the leaders coupl...

To model association fields that underly perceptional organization (gestalt) in psychophysics we consider the problem P (curve) of minimizing for a planar curve having fixed initial and final positions and directions. Here kappa(s) is the curvature of the curve with free total length a"". This problem comes from a model of geometry of vision due to...

We illustrate a multiscale model for social dynamics of large groups. The examples we have in mind span both cases of real dynamics, such as crowd in motion, but also virtual dynamics, such as opinion formation in social networks. The model is based on a measure theoretic approach, where the space of Radon measure with finite mass is endowed with a...

We study the controllability of a closed control-affine quantum system driven by two or more external fields. We provide a sufficient condition for controllability in terms of existence of conical intersections between eigenvalues of the Hamiltonian in dependence of the controls seen as parameters. Such spectral condition is structurally stable in...

Recently, we showed that certain types of polyhedral Lyapunov functions for linear time-invariant systems, are preserved by diagonal Padé approximations, under the assumption that the continuous-time system matrix Ac has distinct eigenvalues. In this technical note, we show that this result also holds true in the case that Ac has non-trivial Jordan...

We first generalize a decomposition of functions on Carnot groups as linear combinations of the Dirac delta and some of its derivatives, where the weights are the moments of the function. We then use the decomposition to describe the large time behavior of solutions of the hypoelliptic heat equation on Carnot groups. The solution is decomposed as a...

We consider the problem of minimizing ∫0ℓ √(ξ2+K2(s)ds) for a planar curve having fixed initial and final positions and directions. The total length ℓ is free. Here s is the variable of arclength parametrization, K(s) is the curvature of the curve and ξ >; 0 a parameter. This problem comes from a model of geometry of vision due to Petitot, Citti an...

In this article, we generalize the Wasserstein distance to measures with different masses. We study the properties of such distance. In particular, we show that it metrizes weak convergence for tight sequences. We use this generalized Wasserstein distance to study a transport equation with source, in which both the vector field and the source depen...

We consider the problem of minimizing $\int_{0}^L \sqrt{\xi^2 +K^2(s)}\, ds $ for a planar curve having fixed initial and final positions and directions. The total length $L$ is free. Here $s$ is the variable of arclength parametrization, $K(s)$ is the curvature of the curve and $\xi>0$ a parameter. This problem comes from a model of geometry of vi...

This technical note has been motivated by the need to assess the preservation of polyhedral Lyapunov functions for stable continuous-time linear systems under numerical discretization of the transition matrix. This problem arises when discretizing linear systems in such a manner as to preserve a certain type of stability of the discrete time approx...

In this paper we show that certain piecewise-linear Lyapunov functions are preserved for LTI systems under Padé approximations. In particular, we present a simple method to find a piecewise-linear Lyapunov function that is so preserved under the Padé discretization of any order and sampling time. This result may be of interest in the discretisation...

Motivated by pedestrian modelling, we study evolution of measures in the Wasserstein space. In particular, we consider the Cauchy problem for a transport equation, where the velocity field depends on the measure itself. We deal with numerical schemes for this problem and prove convergence of a Lagrangian scheme to the solution, when the discretizat...

In this paper we present a model of geometry of vision which generalizes one due to Petitot, Citti and Sarti. One of its main features is that the primary visual cortex V1 lifts the image from \({R}^{2}\) to the bundle of directions of the plane \(PT{\mathbb{R}}^{2} = {\mathbb{R}}^{2} \times{P}^{1}\). Neurons are grouped into orientation columns, e...

In this paper we present a model of geometry of vision which generalizes one due to Petitot, Citti and Sarti. One of its main features is that the primary visual cortex V1 lifts the image from R-2 to the bundle of directions of the plane PTR2 = R-2 x P-1. Neurons are grouped into orientation columns, each of them corresponding to a point of the bun...

In this paper we study a model of geometry of vision due to Petitot, Citti and Sarti. One of the main features of this model is that the primary visual cortex V1 lifts an image from $R^2$ to the bundle of directions of the plane. Neurons are grouped into orientation columns, each of them corresponding to a point of this bundle. In this model a corr...

In this paper we provide explicitly the connection between the hypoelliptic heat kernel for some 3-step sub-Riemannian manifolds and the quartic oscillator. We study the left-invariant sub-Riemannian structure on two nilpotent Lie groups, namely the (2,3,4) group (called the Engel group) and the (2,3,5) group (called the Cartan group or the general...

In this paper we consider the problem of reconstructing a curve that is partially hidden or corrupted by minimizing the functional ¿¿(1+K²)ds, depending both on length and curvature K. We fix starting and ending points as well as initial and final directions. For this functional, we find non-existence of minimizers on various functiona...

We consider the problem of reconstructing a curve that is partially hidden or corrupted by minimizing the functional \( \smallint \sqrt {1 + K_\gamma ^2 ds} \) , depending both on the length and curvature K. We fix starting and ending points as well as initial and final directions. For this functional we discuss the problem of existence of minimize...

We present an invariant definition of the hypoelliptic Laplacian on sub-Riemannian structures with constant growth vector using the Popp's volume form introduced by Montgomery. This definition generalizes the one of the Laplace–Beltrami operator in Riemannian geometry. In the case of left-invariant problems on unimodular Lie groups we prove that it...

In this paper we study the Carnot-Caratheodory metrics on SU(2) ¿ S³, SO(3) and SL(2) induced by their Cartan decomposition and by the Killing form. Besides computing explicitly geodesics and conjugate loci, we compute the cut loci (globally) and we give the expression of the Carnot-Caratheodory distance as the inverse of an elementary...

Fix two points $x,\bar{x}\in S^2$ and two directions (without orientation) $\eta,\bar\eta$ of the velocities in these points. In this paper we are interested to the problem of minimizing the cost $J[\gamma]=\int_0^T \left({\g}_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))+ K^2_{\gamma(t)}{\g}_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t)) \right) ~{\rm d}t$...

We present an invariant definition of the hypoelliptic Laplacian on sub-Riemannian structures with constant growth vector, using the Popp's volume form introduced by Montgomery. This definition generalizes the one of the Laplace-Beltrami operator in Riemannian geometry. In the case of left-invariant problems on unimodular Lie groups we prove that i...