ИСТИНА

This work is devoted to controllability and optimization of vibrations of mechanical systems with distributed parameters. The longitudinal displacements of a thin rectilinear elastic rod are studied. Based on the method of integro-differential relations developed by the authors, a generalized statement of the initial-boundary value problem is proposed, which solution is sought in terms of kinematic and dynamic variables defined in a Sobolev space. The critical time such that the system can be brought to a terminal state is determined for the case of a homogeneous rod controlled by external forces applied at its ends. For fixed time intervals longer than the critical one, the control problem of optimal transfer of the system to the zero state is considered. In this case, the minimized functional has the form of the weighted sum of the mean mechanical energy stored by the rod during its motion and the integral quadratic norm of control functions. By using D’Alembert’s representation, the solution to the direct dynamic problem is derived in the form of traveling waves. By taking into account the properties of the generalized solution, the control problem, which is two-dimensional in space and time, is reduced to a classical one-dimensional quadratic vibrational problem. The latter has a set of traveling waves as unknowns. As a result, the optimal control law and the corresponding motion of the rod are obtained explicitly. Finally, energy characteristics of the optimal motion are analyzed depending on the control time and the weight coefficient in the cost functional.