Boundary kinematic control of a distributed oscillatory system

The problem of the boundary kinematic control of one-dimensional oscillatory systems with distributed parameters in a finite time interval is investigated. The displacement of one of the ends is assumed to be controlled; the other end is assumed to be free. The system is subjected to additional disturbing actions, distributed and concentrated at the ends. The problem is stated by selecting an admissible control to transfer the system from an arbitrary state to the required final state, provided that the integral quadratic functional is minimal. Using the Fourier method, the maximum principle method and the L-problem of moments method, the problem of optimal control is solved explicitly in closed form, and, by d’Alembert's wave propagation method, is represented in terms of the initial functions. The requirements concerning the disturbing actions are established, and also those concerning the initial and final distributions and the duration of the control process, leading to smooth solutions of the problem. The problem of synthesizing the optimal control is discussed. The results are of interest when investigating problems of precision control of mechanical systems possessing considerable elastic compliance (shafts, beams, springs, strings, etc.)