Applications of nonstandard analysis in differentrial game theory

Gordon, Dinah Rose

Abstract

In this study we look at optimal control theory and differential game theory. In the control section, to illustrate some of the nonstandard methods which we will be using, we give existence and uniqueness proofs for standard and Loeb measurable controls. The standard existence is a well-known result, the proof we give is is due to Keisler; this proof was given by him in previously unpublished lecture notes at the University of Wisconsin ([27]). The uniqueness proof is a simple application of Gronwall's Lemma ([31]).

We then show that there is always an optimal Loeb control even in situations where there is no optimal Lebesgue control. Using this result we are then able to show the well known result that there is always a standard optimal relaxed control.

In the games section, by using nonstandard analysis we show that, under certain circumstances, we have the existence of value for two player, zero-sum differential games played over the unit time interval. We follow the work of Elliott and Kalton and, as they did, we show that if the Isaacs condition holds then the game has value in the sense of Friedman. Over the relaxed controls the Isaacs condition is always satisfied and so there is always value for relaxed controls. Like Elliott and Kalton, we do not need Friedman's hypothesis that the variables appear separated in the dynamics and payoff. By using nonstandard methods we are, unlike Elliott and Kalton, able to show these results without using the Isaacs-Bellman equation, other than to explain what the Isaacs condition is. We also find it unnecessary to impose as many restrictions on the functions as Elliott and Kalton.

Citation

Gordon, D. R. (1996). Applications of nonstandard analysis in differentrial game theory. (Thesis). University of Hull. Retrieved from https://hull-repository.worktribe.com/output/4227198

• Thesis Type: Thesis
• Deposit Date: Dec 14, 2016
• Publicly Available Date: Mar 2, 2023
• Keywords: Mathematics
• Public URL: https://hull-repository.worktribe.com/output/4227198
• Additional Information: Department of Mathematics, The University of Hull
• Award Date: Jan 1, 1996