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The equivalence of local and global time-optimal control of a complex of operations. (English) Zbl 0579.49015

Summary: We deal 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 piecewise 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 these 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 optimal control are given.

MSC:

49K15 Optimality conditions for problems involving ordinary differential equations
49J45 Methods involving semicontinuity and convergence; relaxation
93B03 Attainable sets, reachability
93B40 Computational methods in systems theory (MSC2010)
93C15 Control/observation systems governed by ordinary differential equations
65K10 Numerical optimization and variational techniques
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