Sparse Stabilization and Control of the Cucker-Smale Model

From a mathematical point of view self-organization can be described as patterns to which certain dynamical systems modeling social dynamics tend spontaneously to be attracted. In this paper we explore situations beyond self-organization, in particular how to externally control such dynamical systems in order to eventually enforce pattern formation also in those situations where this wished phenomenon does not result from spontaneous convergence. Our focus is on dynamical systems of Cucker-Smale type, modeling consensus emergence, and we question the existence of stabilization and optimal control strategies which require the minimal amount of external intervention for nevertheless inducing consensus in a group of interacting agents. First we follow a greedy approach, by designing instantaneous feedback controls with two dierent sparsity properties: componentwise sparsity, meaning that the controls have at most one nonzero component at every instant of time and their implementation is based on a variational criterion involving ‘1-norm penalization terms; time sparsity, meaning that the number of switchings is bounded on every compact interval of time, and such controls are realized by means of a sample-and-hold procedure. Controls sharing these two sparsity features are very realistic and convenient for practical issues. Moreover we show that among the controls built out of the mentioned variational principle, the maximally sparse ones are instantaneously optimal in terms of the decay rate of a suitably designed Lyapunov functional, measuring the distance from consensus. As a consequence we provide a mathematical justication to the general principle according to which \sparse is better" in the sense that a policy maker, who is not allowed to predict future developments, should always consider more favorable to intervene with stronger action on the fewest possible instantaneous optimal leaders rather than trying to control more agents with minor strength in order to achieve group consensus. We then establish local and global sparse controllability properties to consensus. Finally, we analyze the sparsity of solutions of the nite time optimal control problem where the minimization criterion is a combination of the distance from consensus and of the ‘1-norm of the control. Such an optimization models the situation where the policy maker is actually allowed to observe future developments. We show that the lacunarity of sparsity is related to the codimension of certain manifolds in the space of cotangent vectors.