The equivalence of local and global time-optimal control of a complex of operations

The paper deals with time-optimal control of a complex of independent operations having concave models, when the maximum level of total usage of renewable continuously divisible resource is time-variable and piece-wise constant. In the case considered, neither the moments of a step change of the maximum level of resource, nor their amounts in the consecutive time intervals determined by those moments, are known before starting the control, but they become known in the course of control. An algorithm for the solution of this problem (local optimization of a complex of operations) is proposed. For control determined by this algorithm the performance time of a complex of operations is, in general, greater than the minimum performance time of a complex of operations determined for the same problem, assuming that the maximum level of total usage of resource is at every moment known a priori (global optimization of a complex of operations). The concept of a set of reachable states, and convex analysis, are used to formulate a necessary and sufficient condition that local optimization of a complex of operations ensures a global optimization of this complex. Then, using the theory of function equations a full class of models of operations, for which this condition is satisfied, is determined. These are power models with identical exponents. Some properties of the globally optima! control are given.