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A linear-quadratic (LQ, for short) optimal control problem is considered formean-field stochastic differential equations with constant coefficients in aninfinite horizon. The stabilizability of the control system is studied followedby the discussion of the well-posedness of the LQ problem. The optimal controlcan be expressed as a linear state feedback involving the state and its mean,through the solutions of two algebraic Riccati equations. The solvability ofsuch kind of Riccati equations is investigated by means of semi-definiteprogramming method.

A Linear-Quadratic Optimal Control Problem for Mean-Field Stochastic Differential Equations in Infinite Horizon