Exact controllability of linear mean-field stochastic systems and observability inequality for mean-field backward stochastic differential equations

This paper is concerned with the exact controllability of linear mean-field stochastic systems with time-variant random coefficients. We prove that the exact controllability, the validity of the observability inequality for the dual equation, the unique solvability of a family of optimal control problems, the unique solvability of a family of mean-field forward-backward stochastic differential equations (MF-FBSDEs), and the unique solvability of a family of norm optimal control problems are all equivalent. Therefore, some approaches are provided to investigate the issue of exact controllability.