Social Optimal mean field control problem for population growth model

Abstract

In this paper, we consider a class of social optimal mean field control problem of the population growth model. Suppose a fishpond has N-fish schools, the population of each of them is described by a geometry Brownian motion, the fisherman is on one hand to minimize the total nourishment investments and on another hand to minimize the expected population level measured by the state average of the population of all fish schools. By solving an optimal control problem involved N-controls and approximating the state average appearing in the adjoint equation, a series of decentralized controls is designed which have a property of asymptotic social optimality. Finally, a simulation example is given. KEYWORDS optimal control, mean field, the population growth model, geometry Brownian motion, variation method, fixed point principle MSC CLASSIFICATION 49K45, 93E20

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