Optimal control theory can be seen as bringing together two mathematical disciplines. The first is control theory, which analyzes the properties of systems that can be acted upon through an input (or control). The aim is to take the system from a given initial state to a specified final state, possibly while meeting certain criteria (or constraints). The second is optimization theory, whose purpose is to find, among a collection of solutions, the best solution or solutions according to a chosen optimization criterion.
We solve optimal control problems constantly in our daily lives. Someone driving to work will try to minimize their journey time subject to constraints they have set themselves (making a detour via the supermarket, visiting a friend first…). Here, the system is the person and their vehicle; its dynamics follow from Newton's laws; and the control consists of all the actions the person can take to operate the vehicle (adjusting its speed, changing its trajectory…).
Moving a boat… -------------------
The thesis examines Zermelo-type problems on surfaces of revolution (such as the usual Euclidean sphere S²). These form a particular class of optimal control problems. From the standpoint of optimal control, they are studied using a combination of geometric and numerical methods.
The historical Zermelo problem, formulated in 1931 by the German mathematician Ernst Zermelo, asks for the path that takes a boat from one riverbank to the other in the shortest possible time, under the influence of the current and controlled through its heading angle. The question has since been generalized to higher dimensions, more sophisticated geometric spaces and more general currents. Zermelo-type problems comprise all formulations that can be written mathematically in the same form as the historical problem. They were studied extensively during the 20th century and notably motivated the development of Finsler geometry, which considers Zermelo-type problems on "Riemannian manifolds" (but only for currents described as "weak").