Moscow Mathematical Journal

Volume 16, Issue 3, July–September 2016 pp. 561–598.

Asymptotic Control Theory for a System of Linear Oscillators

Authors: Aleksey Fedorov (1) and Alexander Ovseevich (2)
Author institution:(1) Institute for Problems in Mechanics, Russian Academy of Sciences, 119526, Vernadsky av., 101/1, Moscow, Russia
Russian Quantum Center, 143025 Novaya st. 100, Skolkovo, Moscow, Russia
Laboratoire de Physique Théorique et Modèles Statistiques, CNRS and Université Paris Sud, UMR8626, 91405 Orsay, France
(2) Institute for Problems in Mechanics, Russian Academy of Sciences, 119526, Vernadsky av., 101/1, Moscow, Russia

Summary:

We present an asymptotic control theory for a system of an arbitrary number of linear oscillators under a common bounded control. We suggest a design method of a feedback control for this system. By using the DiPerna–Lions theory of singular ODEs, we prove that the suggested control law correctly defines the motion of the system. The obtained control is asymptotically optimal: the ratio of the motion time to zero under this control to the minimum one is close to 1 if the initial energy of the system is large. The results are partially based on a new perturbation theory of observable linear systems.

2010 Math. Subj. Class. 93B03, 93B07, 93B52.

Keywords: Maximum principle, reachable sets, linear systems.

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