Optimal Control of Longitudinal Motion of an Elastic Rod Using Boundary Forces

This study is devoted to the issues of controllability and optimization of oscillatory motions of dynamic systems with distributed parameters. The longitudinal displacements of a thin rectilinear elastic rod are considered. Based on the method of integrodifferential relations proposed by the authors, a generalized formulation of the initial-boundary value problem is given, the solution of which is sought with respect to the kinematic and dynamic variables in a Sobolev energy space. For the case of a uniform rod controlled by external forces applied at both ends, the critical time for which the system can be brought to the rest is determined and the impossibility for arbitrary initial conditions of bringing the points of the rod to the zero state is shown. For fixed time intervals longer than the critical one, the problem is posed to optimally bring the system to the zero state. In this case, the minimized functional is the mean mechanical energy stored in the rod during motion. It is shown that using the d’Alembert representation (in the form of traveling waves), taking into account the properties of the generalized solution, the two-dimensional in space and time control problem is reduced to the classical one-dimensional quadratic variational problem with fixed ends, which is specified with respect to two unknown d’Alembert functions. Using the methods of the calculus of variations, the optimal control and the corresponding motion of the rod are found explicitly. The dependence of the mean energy stored in the system on the control time is analyzed.