The paper is devoted to a design of a common bounded feedback control
steering a system of an arbitrary number of linear oscillators to the
equilibrium. At high energies, the control is based on the asymptotic theory of
reachable sets of linear systems. With decreasing of the energy, a similar
control with a reduced upper bound is used. On the final stage, the control is
constructed by using the method of common Lyapunov functions. Special attention
is paid to the cases of one and two oscillators.; Comment: 12 pages, 4 figures; presented at the 37th Conference and School on
Information Technologies and Systems (Kaliningrad, September 1-6, 2013); the
subject of the present paper has grown out of study arXiv:1308.6090 (see
Appendix III); published version

We present asymptotical control theory for a system of an arbitrary number of
linear oscillators under common bounded control. We suggest a method for a
design of a feedback control for the system. We prove by using the
DiPerna-Lions theory of singular ODE that the suggested control law correctly
defines a motion of the system. The obtained control is asymptotically optimal:
the ratio of motion time to zero with this control to the minimum one is close
to 1, if the initial energy of the system is large. Some of the results are
based on a new lemma about observability of perturbed autonomous linear
systems.; Comment: 32 pages; v. 2: substantially revised