The optimal quasi-stationary motion of a vibration-driven robot in a viscous medium

We consider a rectilinear quasi-stationary motion of a two-mass system in a viscous medium. The motion of the system as a whole occurs due to periodic movements of the internal mass relatively to the shell. The problem is to describe the law of motion of the internal mass that provides the minimum energy consumption with a specified average velocity of the shell. We propose an algorithm for solving the problem with any law of resistance of the medium. We obtain the energy-optimal law of motion of a spherical shell in a viscous liquid.