On One Problem in Optimal Boundary Control for String Vibrations with a Given Velocity of Points at an Intermediate Moment of Time

This paper considers the problem of optimal boundary control for a string vibration process, for which, in addition to boundary conditions, intermediate conditions are specified. The quality criterion is determined for the entire time interval. The control is realized as a result of displacement of the free end of the string (the other end of the string is assumed to be fixed). We describe a procedure for the equivalent transition to a problem with zero boundary conditions. The constructive algorithm for building optimal control uses the classical moment problem method. Calculation formulas are obtained and implemented for an arbitrary number of first harmonics. The scientific novelty of the work consists in taking into account the intermediate conditions for the rate of change in the string deflection. In addition to the theoretical part, this paper includes an applied section about the results of a computational experiment. It illustrates the use of the formulas obtained and analyzes the effectiveness of the approach proposed. Calculations were performed using MAPLE. The practical significance of the results obtained is ensured by sufficient accuracy which is suitable for applications, and the technique may be in demand when building a boundary control for the string deflection dynamics.