Optimal control of mean-field limit of multiagent systems with and without common noise

We consider a generic, suitable class of optimal control problems under a constraint given by a finite-dimensional SDE-ODE system, describing a system of two interacting species of particles: the herd, described by SDEs, and the herders, described by ODEs with the addition of a control function. In particular, we firstly show that for a low number of herders and for the limit of large number of herd individuals, the SDE-ODE system can be approximated by an infinite-dimensional system given by a McKean-Vlasov single SDE coupled with ODEs. Then, thanks to this we show the $\Gamma-$convergence of the optimal control problem for the finite-dimensional system to a certain optimal control problem for the mean-field system. Differently from Ascione-Castorina-Solombrino [9] (SIAM J. Math. Anal., Vol. 55, No. 6, pp. 6965-6990 (2023)), we do not consider an additive noise for the herd, but a more general class, given by idiosyncratic noises (due to a single herd individual) together with common noise (due to how the environment affects the whole herd), and they are independent one from another. As well as this, we consider a more general class of control functions in the ODEs for herders, where the control is applied not only on the herd dynamics, but also on the herd one.