On the calculation of optimal control for (MIN,+) and (MAX,+) linear systems

Timed event graphs can be represented in Min-plus as well as in Max-plus algebra by a linear state space or transfer function model. In this framework, an optimal control problem can be stated as: given an output trajectory, find the greatest input trajectory which yields an output trajectory not greater than the given trajectory. This problem is usually solved by means of residuation theory using backward equations derived from the state space representation. This paper discuss the optimal control problem and it proposes its calculation by using a recursive equation derived directly from the transfer function of the system.