The Worst Disturbance of an Oscillator with Quadratic-Law Damping by Means of a Force with a Given Integral

The worst disturbance problem for an oscillator with quadratic-law damping is stated. The role of the disturbance is played by an external force applied to the oscillator. It is assumed that this force acts in one direction, that the integral of this force with respect to time is given, and that at the initial instant of time the oscillator is resting in the equilibrium position. It is required to find a time history of the disturbing force that maximizes the maximum (with respect to time) of the absolute value of the displacement of the oscillator’s body from the equilibrium position. The worst disturbance from among rectangular pulses under which the disturbing force is constant on the initial time interval and is equal to zero outside of this interval is found and investigated.