Sampled-data based adaptive mean field games for leader-follower stochastic multi-agent systems

We studied sampled-data based continuous-time linear quadratic Gaussian adaptive mean field games for leader-follower multi-agent systems. Each agent had a dynamical model with the unknown parameters and minimized a quadratic-tracking type cost functional, which was a function of the state average terms of the leaders and followers. We designed the decentralized adaptive control strategies based on the sampled-data, the Nash certainty equivalence principle, and the recursive weighted least squares algorithm. First, for the continuous-time linear quadratic optimal tracking control problem with the unknown parameters of the dynamical model, we proposed an adaptive control law based on the sampled-data and the recursive weighted least squares algorithm and proved that the quadratic-tracking type cost functional under the sampled-data based adaptive control law deviated from that under a conventional full-state based optimal control law with the known parameters of the dynamical model by an infinitesimal of the sample step. Second, for the leader-follower mean field games, by the state aggregation method, we estimated the state average terms of the leaders and followers, respectively. Then, by the estimates of the state average terms and the local sampled-data, we proposed decentralized adaptive control strategies for leaders and followers. Furthermore, we proposed a new two-level Nash equilibrium analytical method and showed that the sequence of the sampled-data based decentralized adaptive control strategies designed was an asymptotic Nash equilibrium as the sample step tends to zero. Finally, we presented a numerical simulation to demonstrate the effectiveness of the sampled-data based decentralized adaptive control strategies.