Controllability Conditions of Distributed Parameter Systems with Small Damping

Problems of controllability and motion planning of distributed parameter systems have been addressed in a number of monographs and research papers (see, e.g., [1-4]). On the one hand, the question of the approximate controllability of a linear time-invariant system on a Hilbert space can be formulated in terms of an invariant subspace of the corresponding adjoint semigroup [4]. On the other hand, the problem of an effective control design remains challenging for a wide range of mechanical systems with distributed parameters. An approach for solving the steering problem was introduced in [8] based on the optimal controls for finite dimensional subsystem. This approach was used in [9] for proving the approximate controllability of the rotating Kirchhoff plate model on an invariant manifold. The goal of this work is to propose a constructive control strategy, based on a reduced model, and to justify that this approach can be used to solve the approximate controllability problem for the whole infinite dimensional system with damping. This paper is organized as follows. In Section 1, the approximate controllability problem is set up and the basic controllability conditions are formulated. These controllability conditions are applied to the system representing infinite dimensional family of oscillators with damping in Section 2. Such a family of oscillators represents a rotating body-beam system.