Mean-field optimal control for biological pattern formation

Abstract

We propose a mean-field optimal control problem for the parameter identification of a given pattern. The cost functional is based on the Wasserstein distance between the probability measures of the modeled and the desired patterns. The first-order optimality conditions corresponding to the optimal control problem are derived using a Lagrangian approach on the mean-field level. Based on these conditions we propose a gradient descent method to identify relevant parameters such as angle of rotation and force scaling which may be spatially inhomogeneous. We discretize the first-order optimality conditions in order to employ the algorithm on the particle level. Moreover, we prove a rate for the convergence of the controls as the number of particles used for the discretization tends to infinity. Numerical results for the spatially homogeneous case demonstrate the feasibility of the approach.

Optimal control problems in transport dynamics with additive noise

“…The latter is a rather effective tool to overcome the curse of dimensionality for systems with a very large number of agents. Indeed, kinetic approximations of multi-agent systems and mean-field optimal control problems, mostly in the deterministic setting, have been proposed in recent literature in connection with a huge number of possible applications, ranging from models for opinion formation [30,51], wealth distribution [29,31,43], traffic or pedestrian flows [2,28,46,47,52], herding problems [1,2,9,20,40,48,54], consensusbased optimization [17,24,33,53] (see also [18,26,38,44,45] for rigorous derivations and further applications and [13,14,15,16,19] for optimality conditions). In the context of multi-agent systems with stochastic noise, but without control, we also refer the reader to [11], while mean-field control problems with diffusion terms have been recently considered in [3,25].…”