How to compute the value of a positional differential game

The present paper deals with the computation of the value of differential games [1-12] for systems described by ordinary differential equations linear in the phase variable with a performance functional explicitly depending on the entire history of the motion. In addition to standard conditions providing the existence of the game value, we only assume that the performance functional is positional [7, p. 41; 8] and has appropriate smoothness properties. Under these assumptions, we present a universal procedure for computing the value function of a game. Just as in [7, pp. 86-97; 9-12], this procedure is based on the backward construction of upper convex hulls of auxiliary functions. It is important that the construction involves convexification only with respect to vectors dual to the phase vector of the original system for given values of an additional scalar parameter. The problem in question arises for the following reasons. It was shown in [11] that the computation of the game value call be reduced to the construction of upper convex hulls psi (j) of appropriate auxiliary functions v in appropriate domains G of the space of dual variables, which comprises the space of vectors m dual to the phase vector of the system and the space of additional parameters dual to adequate information elements of the history of the motion. (Here j = k + 1, k,...,1, where k is a sufficiently large positive integer.) It is also known that, in the case of performance functionals such as the total [7. p. 86; 10] or maximal [7, p. 92] deviation of the motion from a given trajectory (these are typical positional functionals), it indeed suffices to convexify the functions psi (j) only with respect to the vectors m for given values of additional parameters (which substantially increases the computational efficiency). On the other hand, an example illustrating the following fact was given in [11]: if the performance functional; is not a positional functional, then, in general, the convexification of psi (i) with respect to m alone is not sufficient, and one has to perform the convexification jointly with respect to all arguments, including additional parameters. It was shown in [12] that the computation of the game value call be reduced to construction of hulls psi (j) of functioas psi (j) defined (and convexified) in domains G(j) consisting only of vectors m provided that the performance functional is a positional functional of special structure (induced by a family of norms). In the following, we consider the case of a general positional functional.