Optimal Control of a Harmonic Oscillator with Parametric Excitation

Dmitrii Kamzolkin, Victor Ilyutko and Vladimir Ternovski ()
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Dmitrii Kamzolkin: Department of Computational Mathematics and Cybernetics, Shenzhen MSU-BIT University, International University Park Road 1, Shenzhen 518172, China
Victor Ilyutko: Department of Computational Mathematics and Cybernetics, Shenzhen MSU-BIT University, International University Park Road 1, Shenzhen 518172, China
Vladimir Ternovski: Department of Computational Mathematics and Cybernetics, Shenzhen MSU-BIT University, International University Park Road 1, Shenzhen 518172, China

Mathematics, 2024, vol. 12, issue 24, 1-17

Abstract: This paper addresses the time-optimal control problem for a harmonic oscillator characterized by a time-dependent frequency. The primary objective is to determine the minimal time required to transition the system from an initial state, defined by a given position and velocity, to a specified final state, while ensuring that the frequency remains within prescribed bounds. The key challenge lies in identifying the optimal switching times between two available frequencies to meet all boundary conditions efficiently. By examining various boundary scenarios, constructing the reachable set of all admissible trajectories, and employing both analytical techniques and control theory, we develop a robust solution strategy. This work holds particular relevance for practical applications demanding rapid state transitions, such as mechanical vibration control and signal processing, where achieving time-optimal performance is critical. Furthermore, the methods presented are adaptable to a wide range of systems facing similar constraints, providing a versatile and effective framework for time-optimal control.

Keywords: optimal control; harmonic oscillator; Bellman optimality principle; Pontryagin maximum principle (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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