Amplitude Control of Systems’ Oscillations with Friction

A method of searching for the optimal control of the amplitude of one-dimensional oscillations in the vicinity of the equilibrium position is generalized to the case of a scleronomic multidimensional mechanical system with friction. The oscillatory degree of freedom of the system does not lend itself to direct control. Its movement is influenced by other, directly controlled degrees of freedom, whose coordinates are selected as control functions. The number of control functions can include both positional and cyclic coordinates. The method does not use conjugate variables in the sense of L.S. Pontryagin’s maximum principle and does not increase the dimension of the original system of differential equations of motion. The effectiveness of the proposed method is demonstrated using examples of specific oscillatory mechanical models about a pendulum with a support sliding along a cycloid with dry and viscous friction, as well as a six-legged robot from an emergency supine position.