Optimizing length of planar curves

Abstract

This article focuses on the problem of finding a shortest path in plane with obstacles. Problems of such nature occur for instance in robotics or transport and are of great importance. The problem is analyzed using the methods of mathematical analysis and calculus of variations. Definitions of basic concepts of the problem are given. From these definitions, useful properties, such as convexity of the length functional, are proven. These properties are used to show the existence of a solution in one of the considered cases of the problem. Other case of the problem was considered, where it is established under which conditions does a shortest path attain its general form and what this form looks like.