On numerical damping of string vibrations using several stationary actuators

The task is to transfer a string from an initial disturbed state to the rest state in minimum time. The damping of the string vibrations is carried out using several stationary actuators. The functional to be minimized is a certain integral. The damping of vibrations is controlled using a function included in the right-hand side of the hyperbolic equation describing the transverse vibrations of the string and modeling the actions of the actuators. Computational algorithms for solving this problem are developed on the basis of a grid method and the gradient method for finding the minimum of functions of multiple variables with the gradient calculated using the fast automatic differentiation proposed by Yu.G. Yevtushenko. Examples of string damping calculations using different numbers of actuators are given.