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 first show that for a low number of herders and for the limit of a 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 [Formula: see text]-convergence of the optimal control problem for the finite-dimensional system to a certain optimal control problem for the mean-field system. Differently from [ 8 ], 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 of one another. As well as this, we consider a more general class of control functions in the ODEs for herders, where the control depends not only on the herd dynamics but also on the herders’ dynamics.