Optimal trajectories in $L^1$ and under $L^1$ penalizations

Abstract

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 $L^1$ 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. We then use modern algorithms in non-smooth convex optimization in order to obtain a numerical method to simulate such solutions.

Keywords

• BV functions
• non-autonomous calculus of variations
• regularity
• non-smooth optimization