Brachistochrone with limited reaction of constraint in an arbitrary force field

The paper considers brachistochronic motion of a particle along a curve y=y(x) in an arbitrary force field in the vertical plane of Cartesian coordinate system. The curve is treated as a bilateral or unilateral constraint that can be smooth or rough. The projection of the reaction force of the curve onto the normal to the curve is confined to the fixed limits. A control variable u is given as the second derivative of the function y(x) relative to the horizontal coordinate x of the particle, i.e., u=d (2) y/dx (2). Applying Pontryagin's maximum principle and singular optimal control theory, the problem is reduced to numerical solving of the corresponding two-point boundary value problem. The procedure based on the shooting method is used to solve the boundary value problem. Two examples with friction forces of the viscous friction and Coulomb friction type have been solved