A technique for solving a class of differential games†

Abstract

This paper describes a technique for solving a class of two-person zero-sum differential games with complex pay-off functionals, which either cannot be handled by techniques that are presently available or can be handled only with extreme difficulty. The given pay-off functional is decomposed into simpler functionals each of which may be interpreted as a cost scale for the game. The differential game is then solved using a linear combination of these cost scales as the objective functional (hero the use of the term objective functional is intentional in order to avoid the linear combination functional being confused with the original given pay-off functional). The expressions for the optimal controls are obtained in terms of the ratios of the weighting factors in the linear combination objective functional. A search technique is then used either in the cost-scale space or the weighting factors space to determine the optimum value of the ratios of the weighting factors corresponding to the given pay-off functional of the differential game. The application of this technique is illustrated by an example of a linear first-order zero-sum differential game.

Notes

†Communicated by Dr. A. J. Fuller.