Linear Quadratic Optimal Control for Continuous-Time Mean-Field Stochastic Systems with \(\alpha \) -Stability Constraint

This paper is concerned with the continuous-time mean-field linear-quadratic (LQ) optimal control problem on the infinite horizon with \(\alpha \) -stability degree constraint. Based on the condition of exact observability, it is first shown that the considered system is \(\alpha \) -stable if and only if a generalized Lyapunov equation admits a unique positive-definite solution. Moreover, a sufficient condition is given for the solvability of two coupled algebraic Riccati equations. Finally, by the positive definite solutions of coupled algebraic Riccati equations, an optimal control is obtained to reach the minimum of the cost function and to guarantee the convergence rate of the controlled system no slower than the given stability degree.