Title: Mean-Field Sparse Jurdjevic--Quinn Control,
Abstract: We consider nonlinear transport equations with non-local velocity, describing
the time-evolution of a measure, which in practice may represent the density of
a crowd. Such equations often appear by taking the mean-field limit of
finite-dimensional systems modelling collective dynamics. We first give a sense
to dissipativity of these mean-field equations in terms of Lie derivatives of a
Lyapunov function depending on the measure. Then, we address the problem of
controlling such equations by means of a time-varying bounded control action
localized on a time-varying control subset with bounded Lebesgue measure
(sparsity space constraint). Finite-dimensional versions are given by
control-affine systems, which can be stabilized by the well known
Jurdjevic--Quinn procedure. In this paper, assuming that the uncontrolled
dynamics are dissipative, we develop an approach in the spirit of the classical
Jurdjevic--Quinn theorem, showing how to steer the system to an invariant
sublevel of the Lyapunov function. The control function and the control domain
are designed in terms of the Lie derivatives of the Lyapunov function, and
enjoy sparsity properties in the sense that the control support is small.
Finally, we show that our result applies to a large class of kinetic equations
modelling multi-agent dynamics.