On time-optimal control of a sequence of projects of activities under time-variable resource

A sequence of k projects of independent activities, each project composed of activities available for realization at the same time, is considered. It is assumed that the activities are continuous dynamical systems with dynamics that depend continuously on the alloted amounts of the resource and that the initial and terminal states are fixed. The problem is to allocate the time-variable renewable, continuously divisible resource (e.g. power, fuel flow, approximate manpower) to the activities in order to minimize the performance time of the sequence of projects. A solution is presented that is based on the notion of the set of reachable states under the assumption that the allowable level of the total usage of the resource is piecewise constant. Necessary and sufficient optimality conditions are stated in terms of the performance time, and the existence of the optimal control is proved. An algorithm for the time optimal control is proved. An algorithm for the time-optimal resource allocation is derived and an example is given to illustrate the approach