# PDE-constrained optimal control of a leader-follower opinion formation model

## TL;DR

This paper develops a PDE-constrained optimal control framework for leader-follower opinion formation models, deriving associated equations and solving them numerically to analyze different interaction scenarios.

## Contribution

It introduces a novel PDE-constrained optimal control approach for leader-follower opinion dynamics, including derivation of optimality conditions and a numerical solution method.

## Key findings

- Effective numerical algorithms for opinion control.
- Simulation results for various interaction types.
- Insights into influence of control strategies.

## Abstract

We consider the PDE-constrained optimal control of a leader-follower kinetic opinion formation model, with a Fokker-Planck-type system of partial differential equations as a state constraint. We derive the Boltzmann-type and Fokker-Planck-type systems of equations associated with the controlled leader-follower opinion formation model. In a function space setting we derive first-order optimality conditions associated with the PDE-constrained optimal control problem, yielding an optimality system of coupled nonlinear partial differential equations. We employ a gradient-type sweeping algorithm to numerically attack the optimality system obtained from the first-order optimality conditions. We present the results from a finite elements based simulation for different types of interactions and cost functionals.