Sparse kinetic Jurdjevic-Quinn control for mean-field equations

We consider mean-field equations, that are transport partial differential equations describing the evolution of the density of a crowd of interacting agents when their number grows to infinity. We generalize the Jurdjevic-Quinn control strategy to such mean-field equations to achieve stability in a suitable sense. For instance, in the context of crowds, consensus or alignment of agents are among the most relevant goals and they correspond to stability to a suitable manifold. Our main result is the definition of a Jurdjevic-Quinn strategy that is sparse, i.e. that acts on a small set of the configuration space at each time. We show that such a stabilization strategy can be adapted to achieve alignment of the kinetic Cucker-Smale model with a sparse control only.