Linear Quadratic Mean Field Teams: Optimal and Approximately Optimal Decentralized Solutions

We consider team optimal control of decentralized systems with linear dynamics, quadratic costs, and arbitrary disturbance that consist of multiple sub-populations with exchangeable agents (i.e., exchanging two agents within the same sub-population does not affect the dynamics or the cost). Such a system is equivalent to one where the dynamics and costs are coupled across agents through the mean-field (or empirical mean) of the states and actions (even when the primitive random variables are non-exchangeable). Two information structures are investigated. In the first, all agents observe their local state and the mean-field of all sub-populations; in the second, all agents observe their local state but the mean-field of only a subset of the sub-populations. Both information structures are non-classical and not partially nested. Nonetheless, it is shown that linear control strategies are optimal for the first and approximately optimal for the second; the approximation error is inversely proportional to the size of the sub-populations whose mean-fields are not observed. The corresponding gains are determined by the solution of K+1 decoupled standard Riccati equations, where K is the number of sub-populations. The dimensions of the Riccati equations do not depend on the size of the sub-populations; thus the solution complexity is independent of the number of agents. Generalizations to major-minor agents, tracking cost, weighted mean-field, and infinite horizon are provided. The results are illustrated using an example of demand response in smart grids.