A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon

A linear-quadratic (LQ, for short) optimal control problem is consideredfor mean-field stochastic differential equations with constantcoefficients in an infinite horizon. The stabilizability of thecontrol system is studied followed by the discussion of thewell-posedness of the LQ problem. The optimal control can beexpressed as a linear state feedback involving the state and itsmean, through the solutions of two algebraic Riccati equations. Thesolvability of such kind of Riccati equations is investigated bymeans of semi-definite programming method.