Time-Optimal Control Problem of Two Non-Synchronous Oscillators

Leonid Berlin, Andrey Galyaev () and Pavel Lysenko
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Leonid Berlin: Institute of Control Sciences of RAS, 117997 Moscow, Russia
Andrey Galyaev: Institute of Control Sciences of RAS, 117997 Moscow, Russia
Pavel Lysenko: Institute of Control Sciences of RAS, 117997 Moscow, Russia

Mathematics, 2022, vol. 10, issue 19, 1-19

Abstract: The time-optimal control problem for a system consisting of two non-synchronous oscillators is considered. Each oscillator is controlled with a shared limited scalar control. The objective of the control is to accelerate the oscillatory system to a given specific position, where the first oscillator must have non-zero phase coordinates, but the second one must remain motionless at the terminal moment. For an arbitrary number of unknown switching moments that determine the optimal relay control, the necessary extremum conditions in the form of nonlinear matrix equalities are proposed. The study of the necessary/sufficient conditions of the extremum made it possible to describe the reachability set in the phase space of the first oscillator, to find an analytical form of the curve corresponding to the two-switching control class, which also separates the reachability set of the three switching-control class. The corresponding theorems are proved and the dependence of the criteria on control constraints is shown. Matrix conditions for different classes of control switchings are found. All of the obtained analytical results are numerically validated and illustrated with mathematical modeling.

Keywords: optimal control; harmonic oscillator; Pontryagin maximum principle; limited scalar control (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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