An autonomous optimization of oscillations of systems with a control deficit

A scleronomic Lagrangian system with $n$ degrees of freedom is considered, with one degree of freedom being uncontrollable and the rest being controlled by servos. The system is located in the vicinity of equilibrium. A criterion for optimal control of the oscillation amplitude by an uncontrollable degree of freedom due to servo control by other degrees of freedom is proposed. Using examples of specific pendulum systems, the effectiveness of the proposed criterion is demonstrated.