Game theoretic decentralized feedback controls in Markov jump processes

This paper studies a decentralized routing problem over a network through a mean field game approach. Using a state space extension technique we recast the problem in the framework of optimal control. The main contribution is an explicit expression of the optimal decentralized control to provide conditions for the convergence both to local and global equilibrium. Furthermore, we study the stability of the system also in the presence of a delay due to a hysteresis operator. The presence of this operator leads to have more equilibria, nevertheless it is possible to achieve a convergence result. The stability analysis is supported by numerical experiments