Decentralized social-optimal solution of finite number of average field linear quadratic control

Abstract

This paper studies the decentralized solution to the linear quadratic social-optimal mean field control problem, when a finite number subsystems is considered. We use the term average field, to distinguish this case from the one with infinite subsystems. The goal of each subsystem is to design its input to optimize a common cost of social type. To this end, each subsystem use only real-time information which can be either its local state, or a linear measurement noisy measurement of it. Our result generalizes previous ones in three senses. First, it permits designing controls when there are a finite number of subsystems. Second, it permits considering heterogeneous subsystems, that is, having different structural parameters. Third, it permits designing not only state feedback controls but also output feedback ones. In particular, we show that the separation principle holds in the latter case.