Controllability concepts for mean-field dynamics with reduced-rank coefficients
Dan Goreac, Juan Li, Xinru Zhang
TL;DR
The paper addresses exact controllability for mean-field linear SDEs under reduced-rank noise, showing that classical $\mathbb{L}^2$ controllability requires full rank on the diffusion terms, e.g., $\operatorname{rank}(D_1+D_2)=d$ with often $\operatorname{rank}(D_1)=d$. When these conditions fail, the authors derive drift-based rank criteria that enable exact controllability with weaker control regularity, and, if both diffusion-rank conditions fail, they introduce exact terminal controllability to normal laws (ETCNL). They further develop a Wasserstein-set-valued backward SDE framework associated to ETCNL, providing Gaussian-set-valued solution concepts and one-dimensional explicit characterizations. Overall, the work extends mean-field controllability theory to reduced-rank diffusion, connecting exact terminal controllability, Gaussian law reachability, and Wasserstein geometry with potential implications for covariance control and law-based control objectives.
Abstract
In this paper we explore several novel notions of exact controllability for mean-field linear controlled stochastic differential equations (SDEs). 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 $\mathbb{L}^2$-controls necessarily requires both rank conditions on the noise introduced in [8] 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.