Controllability Concepts for Mean-Field Dynamics with Reduced-Rank Coefficients

In this paper we explore several novel notions of exact controllability for mean-field linear controlled stochastic differential equations (MFSDEs). A key feature of our study is that the noise coefficient is not required to be of full rank. We begin by demonstrating that classical exact controllability with [Formula: see text]-controls necessarily requires both rank conditions on the noise introduced in [D. Goreac, Stoch. Anal. Appl., 32 (2014), pp. 280–297] and subsequent works. When these rank conditions are not satisfied, we introduce alternative rank requirements on the drift, which enable exact controllability by relaxing the regularity of the controls. In cases where both the aforementioned rank conditions fail, we propose and characterize a new notion of exact terminal controllability to normal laws (ETCNL). Additionally, we investigate a new class of Wasserstein set-valued backward SDEs that arise naturally associated to ETCNL.