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

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

Financial intermediary distress in the Republic of Korea - Small is beautiful?

Taking the Korean experience as a laboratory experiment in systemic financial crises, the authors analyze distress in individual institutions among two groups of financial intermediaries. They pool together a group of large financial intermediaries (commercial banks, merchant banking corporations) and another group of tiny mutual savings and finance companies. Both the too-big-to-fail doctrine and the credit channel approach suggest that the probability of distress would be greater for small intermediaries. But the authors find that proportionately fewer small intermediaries were distressed than were large intermediaries. They offer two hypothetical explanations for this unexpected result: 1) Exchange rate exposure - a major shock to Korean intermediaries - was presumably negligible for the small financial intermediaries. 2) Small financial intermediaries allocated loans better, because of the"peer monitoring"natural to their mutual nature and deep local roots. Available data did not allow the authors to test the first hypothesis, but they did find support for the second one. Estimating a logit model, they find that the probability of distress was systematically smaller for the mutual savings and finance companies that stayed closer to their origins (for example, collecting many deposits as"credit mutual installment savings") and for those with a longer history of doing business in their local community.Payment Systems&Infrastructure,Banks&Banking Reform,Financial Intermediation,International Terrorism&Counterterrorism,Financial Crisis Management&Restructuring,Financial Crisis Management&Restructuring,Economic Theory&Research,Environmental Economics&Policies,Banks&Banking Reform,Financial Intermediation