Mojtaba Nourian, Roland P. Malhamé, Minyi Huang, and Peter E. Caines
Mean field stochastic control theory, leader–follower collective motion, maximum likelihood ratio
We consider a leader–follower dynamic game model for large population systems where the agents have linear stochastic dynamics and are coupled via their quadratic cost functions. The cost of each leader is based on a tradeoff between moving towards a certain reference trajectory signal and staying near a weighted average of the members’ states. Followers react by tracking the weighted average of the leaders’ states. We approach this large population game problem by use of the so-called mean field (MF), or Nash certainty equivalence (NCE), methodology. First, as for the basic MF (NCE) framework, we show that the set of MF (NCE) control laws for leaders and followers possesses an almost sure єN -Nash equilibrium property for a population of size N where єN goes to zero as N goes to infinity. Second, we consider the case where the leaders track a convex combination of their overall average and a certain reference trajectory signal which is unknown to the followers. The followers use a maximum likelihood estimator on a sample of the leaders’ trajectories to identify the member of a given finite class of models which is generating the reference trajectory of the leaders. It is shown that subject to reasonable conditions the true reference trajectory model is identified in finite time with probability one as the leaders’ population goes to infinity. Simulations for different cases are provided to demonstrate the effectiveness of the model.
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