Diffusion Approximations for Controlled Weakly Interacting Large Finite State Systems with Simultaneous Jumps

TL;DR

This paper studies a large particle system with simultaneous jumps, showing that as the number of particles grows, the system's control problem converges to a diffusion control problem, and near-optimal controls can be constructed.

Contribution

It establishes the convergence of the controlled particle system to a diffusion limit and constructs near-optimal controls based on this limit.

Findings

Value function convergence as N→∞

Existence of near-optimal feedback controls

Numerical experiments demonstrating control policies

Abstract

We consider a rate control problem for an $N$-particle weakly interacting finite state Markov process. The process models the state evolution of a large collection of particles and allows for multiple particles to change state simultaneously. Such models have been proposed for large communication systems (e.g. ad hoc wireless networks) but are also suitable for other settings such as chemical-reaction networks. An associated diffusion control problem is presented and we show that the value function of the $N$-particle controlled system converges to the value function of the limit diffusion control problem as $N\to\infty$. The diffusion coefficient in the limit model is typically degenerate, however under suitable conditions there is an equivalent formulation in terms of a controlled diffusion with a uniformly non-degenerate diffusion coefficient. Using this equivalence, we show that near optimal continuous feedback controls exist for the diffusion control problem. We then construct near asymptotically optimal control policies for the $N$-particle system based on such continuous feedback controls. Results from some numerical experiments are presented.