Sparse stabilization of dynamical systems driven by attraction and avoidance forces

Conditional self-organization and pattern-formation are relevant phenomena arising in biological, social, and economical contexts, and received a growing attention in recent years in mathematical modeling. An important issue related to optimal government strategies is how to design external parsimonious interventions, aiming at enforcing systems to converge to specific patterns. This is in contrast to other models where the players of the systems are allowed to interact freely and are supposed autonomously, either by game rules or by embedded decentralized feedback control rules, to converge to patterns. In this paper we tackle the problem of designing optimal centralized feedback controls for systems of moving particles, subject to mutual attraction and repulsion forces, and friction. Under certain conditions on the attraction and repulsion forces, if the total energy of the system, composed of the sum of its kinetic and potential parts, is below a certain critical threshold, then such systems are known to converge autonomously to the stable configuration of keeping confined and collision avoiding in space, uniformly in time. If the energy is above such a critical level, then the space coherence can be lost. We show that in the latter situation of lost self-organization, one can nevertheless steer the system to return to stable energy levels by feedback controls defined as the minimizers of a certain functional with $l_1$-norm penalty and constraints. Additionally we show that the optimal strategy in this class of controls is necessarily sparse, i.e., the control acts on at most one agent at each time. This is another remarkable example of how homophilious systems, i.e., systems where agents tend to be strongly more influenced by near agents than far ones, are naturally prone to sparse stabilization, explaining the effectiveness of parsimonious interventions of governments in societies.