A multidimensional comparison theorem for SDE with reflection applications to control of a group of independent identical agents

A problem of controlling a group of independent identical agents is considered. Such problems arise in mathematical economics, in robotics (swarms of mini-robots). The model includes a stochastic component. The general method of the Bellman equation [6] cannot be applied to this problem. The solution is found by the comparison method for solutions of SDE using the technique of the Skorokhod reflection operators. The optimal policy admits a realization as a decentralized group control algorithm. The results of the preceding author´s paper [1] are generalized: the state space changes from the line to a half-line with reflection at its base; thus the dynamics of the agent is extended to state dependent. Comparison with partial ordering (in ℝ^d) is a pioneering approach.