Ph.D XXXVIII

Supervisor: Lorenzo Freddi

Phone: 

Room: RIZ1 - L2-07-ND

Mail: pinatto.mattia@spes.uniud.it

Research Project

Optimal control of navigation and application to autonomous navigation

The fundamental mathematical tools of optimal routing and autonomous navigation belong to the field of Mathematical Analysis known as "Optimal Control and Calculus of Variations." The mathematical problems of navigation have been studied since ancient times. For example, Thales' theorem, dating back to the 6th century B.C., is the main geometric tool for determining the interception course between two boats on a nautical chart. More than two thousand years later, Euler’s Law of Sines solves the same problem using the tools of Mathematical Analysis, providing a formula for implementing its resolution on an onboard computer.

In 1931, Ernst Zermelo formulated and solved a fundamental navigation problem (both in aeronautics and maritime contexts) in the presence of a current. In 1937, in the article A Navigation Problem in the Calculus of Variations (American Journal of Mathematics, 59(2):327–334, 1937), E.J. McShane studied the problem of the existence of an optimal control when the speed of the vessel (or aircraft) also depends on time and, most importantly, on the heading direction, as in the case of a sailboat or an aircraft, where descent is faster than ascent.

In 1956, Filippov proved the first general theorem on the existence of solutions for minimum-time control problems (of which Zermelo's and McShane's problems are particular cases), formulating them as a differential inclusion. He required that the control multifunction be upper semicontinuous and that the set of values assumed by the dynamics be compact and convex. The non-convex case was later addressed by Cellina, Ferriero, and Marchini in 2006, and in the last 15 years, hundreds of mathematical studies on navigation problems have been published.

Nevertheless, navigation control problems remain highly complex, and Mathematical Analysis can only solve a small fraction of them. Many of these problems are so intricate that they can currently only be tackled with computational tools, which, however, rely on the equations provided by Mathematical Analysis.

Based on the existing literature, this thesis will primarily focus on studying Zermelo’s problem in the case where the control set (i.e., the set of admissible vehicle speeds) is a strictly convex set, and then extending the study to non-convex sets and the presence of state constraints. Such constraints physically represent areas where navigation is prohibited or obstructed due to obstacles. Based on the obtained results, applications to autonomous navigation will be explored.