A Military Application of Viability: Winning Cones, Differential Inclusions, and Lanchester Type Models for Combat

During the First World War, F.W. Lanchester published his book Aircraft in Warfare: The Dawn of the Fourth Arm[31] in which he proposed several mathematical models based on differential equations to describe combat situations. Since then, his work has been extensively modified to represent a variety of competitions, ranging from isolated battles to entire wars. There exists a class of mathematical models known under the name of differential Lanchester type models. Such models have been studied from different points of view by many authors in hundreds of papers and unpublished reports. We note that Lanchester type models are used in the planning of optimal strategies, supply, and tactics. In our first paper on the subject [27], we studied Lanchester type models from a viability standpoint through the introduction of the new notion of winning cone. We have also considered a variation on optimal control that we call Optimal Control by Viability. Although the subject was mentioned, the difficulties and well-known problems associated with Lanchester coefficients was not considered in this first part. Herein, we turn our attention to these coefficients and, to overcome this obstacle and facilitate the application of such models, we will introduce the notion of Lanchester type differential inclusionsthrough the replacement of the classic coeffi- cients by intervals. We will show how viability theory for set-valued mappings can be applied to determine viability conditions for the winning cone. In the last section, we will again consider Optimal Control by Viability, but in the set-valued case represented by differential inclusions.