Approximately global optimal control for max-plus linear systems and its application on load distribution

This paper considers the approximation problem of global optimal control of max-plus linear systems with real affine equality constrains. The approximate model that makes feasible values closest to the greatest lower bound is set up by introducing the distance between max-plus vectors. The existence of approximately optimal solutions is proved, and a criterion for the uniqueness of such solutions is presented. A general formula with polynomial time complexity is given to compute all the approximately optimal solutions. Furthermore, the proposed approximation technique is used to minimise the overall completion time of a distributed system without accurate optimal solution. The examples and simulations are provided to demonstrate the applicability of the results.