Integral minimization of the second derivative module of the boundary control of string vibrations with the fixed end

A problem of boundary control of string vibrations with the fixed end with a bounded control resource is studied. The integral of the second derivative module of the boundary control is minimized. Estimates of upper and lower bounds are found for the extremum. In the vibration suppression problem, at the time period $T$ formulas of boundary control that is close to the optimal control with large values of the time period $T$ are given.