Optimization of a boundary control by a displacement at one end of a string with second end free during an arbitrary sufficiently large time interval

Optimal singular and chattering modes in the problem of controlling the vibrations of a string with clamped ends

“…Here the control is either a displacement of one of the ends of the string, 1 or an elastic force acting on one of the ends, 2 or a combined control is considered, namely, an elastic force, applied to the left end of the string and a displacement of the right end. 3 The problem of the transfer of the string from a rest position to a specified fixed position in a specified time, when the displacement of one of the ends of the string serves as the control was considered in Ref.…”

“…The problem of controlling the vibrations of a string was considered previously [1][2][3] in the class of generalized solutions, and a method of determining the optimal control was presented for which the string transfers from an arbitrary specified position to an arbitrary specified final position with a minimum value of the boundary energy integral. Here the control is either a displacement of one of the ends of the string, 1 or an elastic force acting on one of the ends, 2 or a combined control is considered, namely, an elastic force, applied to the left end of the string and a displacement of the right end.…”