Abstract
A reciprocal or dual problem is defined and a reciprocity theorem is derived from the optimal control problem of nonlinear large scale dynamical systems. It is particularly emphasized that the problem is solved by the method of decomposition technique.
When the primal problem is defined as to minimize the specified objective function under the given constraints, the reciprocal or dual problem is defined as to maximize a function constructed from both the original objective function and the constraints, under the Euler and Clebsch equations as the constraints.
Two kinds of reciprocal problems are derived, depending upon convexity conditions of functions involved in the original problems.
The reciprocity theorem assures that the values of objective functions of the primal and the reciprocal problems constitute the upper and lower bounds respectively of the estimated range of the optimal value of the objective function of the primal problem when they are computed along a nominal admissible trajectry and, further, that they coincide with each other when they reach the optimum.
This property of the reciprocal problem seems to be very beneficial in solving the optimal control problem by mean of successive computation, since the values of the objective functions of the both problems give knowledges about the degree of convergence and the value of the further possible decrease of the objective function. This is good particularly when a feasible method of computation in the decentralized optimization is employed.
