Dr. Giacomo Ascione “Well-posedness of nonlinear nonlocal hyper-parabolic Kolmogorov-Fokker-Planck-type PDEs“

In recent years a lot of attention has been given to the control of multiagent leader-follower systems. In particular, in the herding problem, one assumes that such a control can be only applied on the dynamics of the leaders. However, in several situations, such as, for example, in crowd control, while there could be not so many leaders, one can have a huge number of followers. In such a case, the problem is impracticable even with a numerical approach. A possible approximation procedure consists in assuming that one has an infinite number of followers: in such a case, under some assumptions on the interaction terms, the dynamic converges (in some sense) towards a new system, called the mean field limit. In our case, we will assume that, while the dynamics of the leaders are described by means of second order ordinary differential equations (ODEs), the velocity of the followers evolves according to a stochastic differential equation (SDE) with additive noise. Hence, the mean field system, in terms of the dynamics of the prototypical agents, is
given by a system of ODEs (for the leaders) and a McKean-Vlasov equation describing the dynamic of the generic follower. On the other hand, concerning the Fokker-Plank equation, we get a system of as many ODEs as the leaders and a single nonlinear (space-)nonlocal hyper-parabolic partial differential equation (PDE). To study the well-posedness of the system, we adopt a fixed point argument, in which, however, we have to incorporate the ODEs into the PDE. This adds to the aforementioned PDE also a time-nonlocal dependence. For this reason, in this talk, once the main motivations have been established, we will give some well-posedness results for a family of nonlinear (time- and space-)nonlocal hyper-parabolic PDEs. To do this, first we need to address this problem for a linear, local hyper-parabolic PDE with unbounded drift. Once this has been done, well-posedness of the nonlinear nonlocal case will follow by means of a fixed point argument. If there’s time we will discuss the application of such results to the theory of the sparse optimal control of a mean field system as described before.

This is part of a joint work with Francesca Anceschi, from Università Politecnica delle Marche, Daniele Castorina and Francesco Solombrino, from Università degli Studi
di Napoli Federico II.