Sparse Control of Alignment Models in High Dimension

For high dimensional particle systems, governed by smooth nonlinearities depending on mutual distances between particles, one can construct low-dimensional representations of the dynamical system, which allow the learning of nearly optimal control strategies in high dimension with overwhelming confidence. In this paper we present an instance of this general statement tailored to the sparse control of models of consensus emergence in high dimension, projected to lower dimensions by means of random linear maps. We show that one can steer, nearly optimally and with high probability, a high-dimensional alignment model to consensus by acting at each switching time on one agent of the system only, with a control rule chosen essentially exclusively according to information gathered from a randomly drawn low-dimensional representation of the control system.Comment: 39 page

Mean-Field Pontryagin Maximum Principle

International audienceWe derive a Maximum Principle for optimal control problems with constraints given by the coupling of a system of ordinary differential equations and a partial differential equation of Vlasov-type with smooth interaction kernel. Such problems arise naturally as Gamma-limits of optimal control problems constrained by ordinary differential equations, modeling, for instance, external interventions on crowd dynamics by means of leaders. We obtain these first-order optimality conditions in the form of Hamiltonian flows in the Wasserstein space of probability measures with forward-backward boundary conditions with respect to the first and second marginals, respectively. In particular, we recover the equations and their solutions by means of a constructive procedure, which can be seen as the mean-field limit of the Pontryagin Maximum Principle applied to the optimal control problem for the discretized density, under a suitable scaling of the adjoint variables

Optimal control problems in transport dynamics

In the present paper we deal with an optimal control problem related to a model in population dynamics; more precisely, the goal is to modify the behavior of a given density of individuals via another population of agents interacting with the first. The cost functional to be minimized to determine the dynamics of the second population takes into account the desired target or configuration to be reached as well as the quantity of control agents. Several applications may fall into this framework, as for instance driving a mass of pedestrian in (or out of) a certain location; influencing the stock market by acting on a small quantity of key investors; controlling a swarm of unmanned aerial vehicles by means of few piloted drones

Invisible control of self-organizing agents leaving unknown environments

In this paper we are concerned with multiscale modeling, control, and simulation of self-organizing agents leaving an unknown area under limited visibility, with special emphasis on crowds. We first introduce a new microscopic model characterized by an exploration phase and an evacuation phase. The main ingredients of the model are an alignment term, accounting for the herding effect typical of uncertain behavior, and a random walk, accounting for the need to explore the environment under limited visibility. We consider both metrical and topological interactions. Moreover, a few special agents, the leaders, not recognized as such by the crowd, are "hidden" in the crowd with a special controlled dynamics. Next, relying on a Boltzmann approach, we derive a mesoscopic model for a continuum density of followers, coupled with a microscopic description for the leaders' dynamics. Finally, optimal control of the crowd is studied. It is assumed that leaders exploit the herding effect in order to steer the crowd towards the exits and reduce clogging. Locally-optimal behavior of leaders is computed. Numerical simulations show the efficiency of the optimization methods in both microscopic and mesoscopic settings. We also perform a real experiment with people to study the feasibility of the proposed bottom-up crowd control technique.Comment: in SIAM J. Appl. Math, 201

Mean Field Games of Controls with Dirichlet boundary conditions

In this paper we study a mean-field games system with Dirichlet boundary conditions in a closed domain and in a mean-field of control setting, that is in which the dynamics of each agent is affected not only by the average position of the rest of the agents but also by their average optimal choice. This setting allows the modeling of more realistic real-life scenarios in which agents not only will leave the domain at a certain point in time (like during the evacuation of pedestrians or in debt refinancing dynamics) but also act competitively to anticipate the strategies of the other agents. We shall establish the existence of Nash Equilibria for such class of mean-field of controls systems under certain regularity assumptions on the dynamics and the Lagrangian cost. Much of the paper is devoted to establishing several a priori estimates which are needed to circumvent the fact that the mass is not conserved (as we are in a Dirichlet boundary condition setting). In the conclusive sections, we provide examples of systems falling into our framework as well as numerical implementations

Partial adjustment in policy functions of structural models of capital structure

Treball fi de màster de: Master's Degree in Economics and FinanceDirector: Filippo Ippolito
We present a tradeoff model of capital structure to investigate the sources of adjustment costs and study how firms' financing decisions determine partial adjustment toward target leverage ratios. The presence of market imperfections, like taxes and collateral constraints, is shown to play a decisive role in the behavior of the policy function of capital and leverage. By means of a contraction argument, we are able to show the existence of a target leverage towards which optimal leverage converges with a speed of adjustment that depends on a firm marginal productivity of capital. Our predictions are consistent with the empirical literature regarding both the magnitude of the speed of adjustment and the relationship between leverage ratios and the business cycle.Presentem un model de compensació d'estructura de capital per investigar les fonts dels costos d'ajustament i estudiem com les decisions de finançament de les empreses determinen un ajust parcial a les ràtios d'apalancament objectiu. La presència d'imperfeccions del mercat, com ara impostos i restriccions de garantia, demostra un paper decisiu en el comportament de la funció política del capital i l'apalancament. Mitjançant un argument de contracció, podem demostrar l'existència d'un apalancament objectiu cap al qual convergeix un palanquejament òptim amb una velocitat d'ajustament que depèn d'una productivitat marginal ferma del capital. Les nostres prediccions són coherents amb la literatura empírica quant a la magnitud de la velocitat d'ajust i la relació entre els coeficients d'apalancament i el cicle econòmic

(UN)conditional consensus emergence under perturbed and decentralized feedback controls

We study the problem of consensus emergence in multi-agent systems via external feedback controllers. We consider a set of agents interacting with dynamics given by a Cucker-Smale type of model, and study its consensus stabilization by means of centralized and decentralized control configurations. We present a characterization of consensus emergence for systems with different feedback structures, such as leader-based configurations, perturbed information feedback, and feedback computed upon spatially confined information. We characterize consensus emergence for this latter design as a parameter-dependent transition regime between self-regulation and centralized feedback stabilization. Numerical experiments illustrate the different features of the proposed designs