Sparse control of alignment models in high dimension

Abstract

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.

Mean-Field Pontryagin Maximum Principle

“…However, in reality, societies exhibit either convergence to undesired patterns or tendencies toward instability that only an external government can successfully dominate. The need of such interventions, together with the limited amount of resources that governments have at their disposal, makes the design of parsimonious stabilization strategies a key issue, which has been extensively studied in the context of dynamics given by systems of ODEs, see [7,8,9,10,13].…”

Invisible Control of Self-Organizing Agents Leaving Unknown Environments

“…This is the most effective but also the most "expensive" control technique because of the large number of interactions with the agents that is required [13]. A more parsimonious control technique, called sparse control, can be obtained by penalizing the number of these interventions by means of an L 1 control cost, so that the control is active only on few agents at every instant [11,12,18]. If the existing behavioral rules of the agents cannot be redesigned, the control of the system can be obtained by means of external agents.…”

Sparse Control of Multiagent Systems

“…This means that B distinguishes two vectors modulo their projection on V f . Moreover, from (10) immediately follows that B restricted to V ⊥ × V ⊥ coincides, up to a factor 1/N, with the usual scalar product on R dN .…”

“…The situation where the sparse control strategy works at its bests is when the velocities of the agents are almost homogeneous, except for few outliers which are very distant from the mean velocity. As extensively discussed in [10], in such situations the total control is suboptimal because it also acts on agents which do not need any intervention, while the sparse control strategy is locally optimal because it focuses all its strength on the small group of outliers. Such scenario is portrayed in Figure 14: starting from the same initial datum of Figure 13, we modify the velocity of one agent so that it decisively deviates from the mean velocity.…”