Abstract
The classical optimization problem in function space, expressed in simple form leads to the formulation of the auxiliary conditions derived from the maximum principle. Among the methods which use a computer to solve this problem, most consist in obtaining numerical solutions of the differential equations (2) and (5) by treating them as initial value problems -However, boundary conditions for the system (2) are generally specified at the initial time t=t, whilst those of the ad joint system (5) are generally specified at the final time t=t, The numerical solution of those differential equations by initial value techniques would therefore suggest that (2) be integrated in the forward direction (from t=t toward t=t 1) whilst the ad joint system (5) be integrated in the backward direction, (i.e. from t=t 1 towards t=t,). In that case, the boundary conditions are easily satisfied, but the satisfaction of the coupling between the 2 systems of equations (2) and (5) must be obtained by iteration, as for instance in Bryson's "gradient method" (ref. [6], [7]). Alternatively, the "neighboring extremal" or "indirect" methods consist in numerically integrating in the forward direction simultaneously both systems of equations. In this case, the coupling between the 2 systems is always satisfied, but the terminal conditions (in t=t must be satisfied by iteration. As in any problem which involves the numerical solution of ordinary differential equations, one must be ensured of stability in the propagation of computational errors (not to be confused with numerical stability per se) -Because of this requirement, the choice between the two alternatives of numerical implementation described above is not free, since, as we will show, stability in the propagation of errors for a system of differential equations is directly related to the direction in which the integration is being performed. A further analysis will reveal that computing errors propagation stability properties for the system (2) and its ad joint (5) are somehow complementary. Analyzing this problem in some detail and drawing appropriate conclusions are the objectives of this text.