Time-optimal crossing of a sphere in a viscous medium

Abstract

Multidimensional controlled motions of a material point in a homogeneous viscous medium are considered. The problem of steering this point to a fixed sphere (from the outside or from the inside) for a minimum time by a force with a bounded absolute value is solved. For an arbitrary initial position and any initial velocity, optimal control both in the open-loop form of the program and in the feedback form, optimal time and Bellman function, as well as optimal phase trajectory are constructed in an explicit form using the Pontryagin’s maximum principle. The solution is studied analytically and numerically, and qualitative mechanical properties of optimal characteristics of motion are found (nonmonotonic dependence of optimal time on the value of initial vector of velocity, discontinuity of the Bellman function, and so on).