This book should be considered as an introduction to a special dass of
hierarchical systems of optimal control, where subsystems are described
by partial differential equations of various types. Optimization is
carried out by means of a two-level scheme, where the center optimizes
coordination for the upper level and subsystems find the optimal
solutions for independent local problems. The main algorithm is a method
of iterative aggregation. The coordinator solves the problern with
macrovariables, whose number is less than the number of initial
variables. This problern is often very simple. On the lower level, we
have the usual optimal control problems of math- ematical physics, which
are far simpler than the initial statements. Thus, the decomposition (or
reduction to problems ofless dimensions) is obtained. The algorithm
constructs a sequence of so-called disaggregated solutions that are
feasible for the main problern and converge to its optimal solutionunder
certain assumptions ( e.g., under strict convexity of the input
functions). Thus, we bridge the gap between two disciplines:
optimization theory of large-scale systems and mathematical physics. The
first motivation was a special model of branch planning, where the final
product obeys a preset assortment relation. The ratio coefficient is
maximized. Constraints are given in the form of linear inequalities with
block diagonal structure of the part of a matrix that corresponds to
subsystems. The central coordinator assem- bles the final production
from the components produced by the subsystems.
