Differential inclusions, viability and optimal control: An application to Lanchester type models

During the first World War, F.W. Lanchester published his book Aircraft in Warfare: The Dawn of the Fourth Arm [138] 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 Such models have been studied from different point of views by many authors in hundreds of papers and unpublished reports. Note that Lanchester type models are used in the planning of optimal strategies, supply and tactics.; This thesis shows how differential Lanchester type models can be studied from a viability theory stand point. In this sense, the notion of winning cone is introduced and it is shown that for many models, the winning cone is a viable cone.; About the applicability of Lanchester type models, it is known that the estimation of the Lanchester coefficients is a big problem. To overcome this obstacle and facilitate the application of such models, the notion of Lanchester type differential inclusions is introduced through the replacement of the classical coefficients by intervals. In this setting, the following problem is considered: Under what conditions is the winning cone also a viable cone?; In the last part of this thesis, it is shown how the viability theory of differential equations can be used to study the Lanchester type models from the optimal theory point of view.