Mean-field optimal control for biological pattern formation

Martin Burger, Lisa Maria Kreusser, Claudia Totzeck

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Article number40
JournalESAIM - Control, Optimisation and Calculus of Variations
Publication statusPublished - 30 Apr 2021


  • Dynamical systems
  • Interacting particle systems
  • Mean-field limits
  • Optimal control with ODE/PDE constraints
  • Pattern formation

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Control and Optimization
  • Computational Mathematics

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