Minimal norm control problems for linear mean-field control systems

This paper studies controllability and minimal norm control problems (NP problems) for mean-field stochastic differential equation (MF-SDE) systems. We first establish an equivalent rank criterion for terminal controllability, namely, that the diffusion matrices $D$ and $D+\bar{D}$ are both of full row rank. We then derive an explicit Riccati-based representation of the minimal norm control by applying a parameterization technique. A key step is to identify the value $P(0)$ of an associated Riccati solution with the controllability Gramian of a controllable auxiliary system, which yields the parameter required in the construction. The resulting representation provides a basis for the future development and analysis of numerical methods for NP problems.