Filippo Santambrogio

Institut Camille Jordan, Université Claude Bernard - Lyon 1

Optimal  trajectories  in L1 and under L1 penalizations

I will present a recent work in collaboration with my student Annette Dumas. Motivated by a  MFG  model where the  trajectories  of the agents are piecewise constants and agents pay for the number of jumps, we study a variational problem for curves of measures where the cost includes the length of the curve measures with the L1 distance, as well as other, non-autonomous, cost terms arising from congestion effects.

We prove several regularity results (first in time, then in space) on the solution, based on suitable approximation and maximum principle techniques. Also the finite dimensional case, where trajectories  are just valued into Rn but their  velocity is penalized through its L1 norm, is interesting.