An optimal control problem for linear SDE of mean-field type with terminal constraint and partial information

This paper is concerned with an optimal control problem for a linear stochastic differential equation (SDE) of mean-field type, where the drift coefficient of observation equation is linear with respect to the state, the control and their expectations, and the state is subject to a terminal constraint. The control problem cannot be solved by transforming it into a standard optimal control problem for an SDE without mean-field term. By virtue of a backward separation method with a decomposition technique, one optimality condition and one forward-backward filter are derived. Two linear-quadratic (LQ) optimal control problems and one cash management problem with terminal constraint and partial information are studied, and optimal feedback controls are explicitly obtained.