Some Remarks on LQ Mean-Field Social Control Problems for Stochastic Input Delay Systems

In this letter, we consider linear-quadratic (LQ) mean-field social control problems for a class of stochastic systems with ordinary control input and delay control input. We define a stabilization problem via a memoryless static output feedback (SOF) strategy and then solve the problem of minimizing the upper bound of the cost function using guaranteed cost control theory. It is found that the minimization of the upper bound of the cost function cannot be attained if only a delay control input exists. Furthermore, it is proved that it is impossible to implement a mean-field SOF strategy to solve the minimization problem, and the input matrix must have the same dimension as the state matrix. To solve this minimization problem, the necessary conditions for the sub-optimality are established via stochastic cross-coupled matrix equations (SCCMEs) using the Karush-Kuhn-Tucker condition and the state feedback strategy. Finally, the performance and usefulness of the proposed strategy are investigated using an order-reduced scheme based on the direct method.