期刊
MATHEMATICS
卷 10, 期 19, 页码 -出版社
MDPI
DOI: 10.3390/math10193552
关键词
optimal control; harmonic oscillator; Pontryagin maximum principle; limited scalar control
类别
资金
- grant to support youth scientific schools of ICS RAS Methods for trajectories optimization of controlled objects
The time-optimal control problem for a system consisting of two non-synchronous oscillators is examined in this study. By proposing necessary extremum conditions in the form of nonlinear matrix equalities, the relationship between the reachability set and control classes of the oscillators is described. The obtained analytical results are validated through mathematical modeling.
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.
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