On the control of density-dependent stochastic population processes with time-varying behavior

The study of density-dependent stochastic population processes (DDSPPs) is important from a historical perspective as well as from the perspective of a number of existing and emerging applications today. In more recent applications of these processes, it can be especially important to include time-varying parameters for the rates that impact the density-dependent population structures and behaviors. Under a mean-field scaling, we show that such time-inhomogeneous DDSPPs converge to a corresponding nonautonomous dynamical system. We then analogously establish that the optimal control of such time-inhomogeneous DDSPPs converges to the optimal control of the limiting dynamical system. An analysis of both the dynamical system and its optimal control renders various important mathematical properties of interest.