A linear quadratic control problem with mean field dependent fixed costs

The paper deals with an optimal control problem in one dimension, having affine dynamics, and a running cost which is discontinuous in the control variable. More precisely, besides terms which are quadratic in the state and control variables, a mean field dependent fixed cost is paid as long as the control is activated (not null). This leads to a problem that, although simple, does not seem to fall into a known class. Moreover, the outcome is completely different according to the magnitude of the fixed cost in comparison to other parameters, such as the (constant) disturbance appearing into the state equation. By means of some tools of Bellman's Dynamic Programming and viscosity solutions, we are able to provide an explicit formula for the value function in all different cases, as well as a feedback formula for the optimal control, leading in some subcases to chattering optimal controls. Finally a mean field game is introduced where a set of homogeneous players minimize each the cost functional of the single player control problem, sharing the fixed cost. For such a game the existence of equilibrium points is proved and their characterization is displayed.