Abstract. A general class of mean field games are considered where the governing dynamics are controlled diffusions in R d . The optimization criterion is the long time average of a running cost function. Under various sets of hypotheses, we establish the existence of mean field game solutions. We also study the long time behavior of the mean field game solutions associated with the finite horizon problem, and under the assumption of geometric ergodicity for the dynamics, we show that these converge to the ergodic mean field game solution as the horizon tends to infinity. Lastly, we study the associated N -player games, show existence of Nash equilibria, and establish the convergence of the solutions associated to Nash equilibria of the game to a mean field game solution as N → ∞.
Abstract:The electrical system on a ship is well contained. Among the challenges proffered by naval systems is real time monitoring for the purpose of fault analysis and reconfiguration. The power system can be considered as a grid of interconnected trunk lines each with its own equivalent parallel load impedance and series transmission impedance. These equivalent impedances provide a natural means of reducing a rather large, complex system to a small compact system. The reduced system can be optimized to maximize power flow through the equivalent parallel impedances and minimize the loss in the series components. Binary integer optimization techniques offer the most promise in solving these problems quickly.
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