Optimal switching between collective motion states for two agents

We show that steering control can be chosen to give bistability between parallel and anti-parallel collective motion states for a continuous-time kinetic model of two agents moving in the plane with unit speed. Variational methods are used to determine the optimal input to the steering control of one of the agents which leads to switching between these collective states. For any given time interval of switching, such an optimal input is shown to exist and to be unique. The properties of optimal inputs are interpreted by considering the phase space geometry of the Euler–Lagrange equations associated with the optimization.