Game Theoretic Decentralized Feedback Controls in Markov Jump Processes

Abstract

This paper studies a decentralized routing problem over a network, using the paradigm of mean-field games with large number of players. Building on a state space extension technique, we turn the problem into an optimal control one for each single player. The main contribution is an explicit expression of the optimal decentralized control which guarantees the convergence both to local and global equilibrium points. Furthermore, we study the stability of the system also in the presence of a delay which we model using an hysteresis operator. As a result of the hysteresis, we prove existence of multiple equilibrium points and analyze convergence conditions. The stability of the system is illustrated via numerical studies .

A mean field approach to model flows of agents with path preferences over a network

“…Notice that an ε-optimal stream associated to a general partition τ may or may not exist, but it certainly does when the partition is refined enough. Indeed, considering the functions involved in the minimization process in (16), (18), (20), (22), if the minimum is realized up to the error ε, the minima are attained within intervals of type [t + h ε , T ], for some suitable h ε > 0, so that the functions cited above are Lipschitz continuous. Denote by L the greater of Lipschitz constants of these functions.…”

Hysteresis and controllability of affine driftless systems: some case studies

“…On one hand we apply it in the controls (see (1.2)-left), on the other hand we introduce the hysteresis in the state variables (1.2)-right. These two cases may model respectively the situation where the control is performed by an external magnetic field (see for example Alouges et al [2,3]) and where there could be a sort of lack of information in the state-variable, for example in the synthesis of feedback controls (see for example Logemann et al [25], Cocetti et al [15] and Bagagiolo et al [8], and Tarbouriech et al [31] for the case of linear systems). The first case is addressed in Section 3.…”

Hysteresis and controllability of affine driftless systems: some case studie

“…These two cases may model respectively the situation where the control is performed by an external magnetic field (see for example Alouges at al. [2,3]) and where there could be a sort of lack of information in the state-variable, for example in the synthesis of feedback controls (see for example Bauso et al [6], Cocetti et al [14], Logemann et al [26] and Tarbouriech et al [30] for the case of linear systems). The first case is addressed in Section 2.…”