Brachistochronic motion of a nonholonomic variable

In this paper, the brachistochronic motion of a mechanical system composed of variable-mass particles is analysed. Workless (ideal) holonomic and linear nonholonomic constraints are imposed on the system. It is assumed that the system moves in an arbitrary field of known potential and nonpotential forces with prescribed both laws of the time-rate of mass variation of the particles and relative velocities of the expelled (or gained) masses. The first time-derivatives of quasi-velocities are taken as control variables. Using Pontryagin’s maximum principle and singular optimal control theory, the problem of brachistochronic motion of the nonholonomic variable-mass mechanical system is solved as a two-point boundary value problem. In addition, a discussion about the realization of control forces is given. The results are illustrated via an example