Optimal control and stabilization for linear mean-field system with indefinite quadratic cost functional

In this paper, we consider an optimal control and stabilization problem of linear mean-field (MF) system, where the quadratic cost functional is allowed to be indefinite. Inspired by the equivalent cost functional method, we introduce a subset, which helps us to investigate the convergence property of generalized differential Riccati equations arising in indefinite mean-field linear quadratic (MF-LQ) problems with finite horizon. A coupled generalized algebraic Riccati equation (GARE) is thus obtained. More importantly, the solution pair of corresponding GARE can be decomposed into a solution pair of a coupled standard algebraic Riccati equation (SARE) and a matrix pair in the subset we introduce. Thus, an equivalent relationship is established between GARE and SARE. With this equivalence, we derive necessary and sufficient conditions to stabilize linear MF-system with indefinite weighting matrices in mean-square sense.