Brachistochrone with coulomb friction

This paper formulates and solves in closed form the problem of finding the minimum-time path of a particle between two points in a uniform gravitational field when motion of the particle is resisted by a force proportional to the normal force exerted on the particle by the path. This resistance to motion is the common mathematical form for Coulomb friction. The problem solution involves the reformulation of the classical brachistochrone of Bernoulli in terms of a singular control problem in which the time derivative of the heading angle of the particle is the control parameter. As such, this solution provides a unique approach to the solution of minimum-time path problems.