The Fastest Damping of a Linear Inverted Pendulum Using a Dynamic Absorber

We consider a linear controllable third-order mechanical system whose matrix has one positive eigenvalue. The system can serve as a mathematical model of a linearized inverse pendulum controlled by an active dynamic damper. A modulus constraint is imposed on the value of the control variable. Using Pontryagin’s maximum principle, the problem of synthesizing an optimal control that brings the system to a state of rest in a minimum time is solved.