Social Optima in Mean Field Linear-Quadratic-Gaussian Models with Markov Jump Parameters

This paper investigates social optima of mean field linear-quadratic-Gaussian (LQG) control models with Markov jump parameters. The common objective of the agents is to minimize a social cost—the cost average of the whole society. In the cost functions there are coupled mean field terms. First, we consider the centralized case and get a parameterized equation of mean field effect. Then, we design a set of distributed strategies by solving a limiting optimal control problem in an augmented state space subject to the consistency requirement for mean field approximation. It is shown that the set of distributed strategies is asymptotically team-optimal, and the asymptotically optimal social cost value can be obtained explicitly. The optimal social average cost is compared with the optimal individual cost in mean field games by virtue of the explicit expressions, and the difference is further illustrated by a numerical example.