Optimal boundary control of string vibrations with intermediate values of the deflection function under minimization of the boundary energy integral

The paper studies optimal boundary control problems with given intermediate values of the deflection function for the string vibration equation with given initial and final conditions. The boundary energy integral serves as the quality criterion. The controls are either a displacement at one end with the other end fixed or a displacement at both ends of the string. The quality criterion is set over the entire time interval. We rely on the methods of separation of variables and optimal control theory with multipoint intermediate conditions to propose a constructive approach for deriving an optimal boundary control of string vibrations. We also perform a computational experiment and analyze its results.