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Numerical approximation of constrained optimal control problems in delayed systems using an enhanced Rayleigh-Ritz algorithm.

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Title: Numerical approximation of constrained optimal control problems in delayed systems using an enhanced Rayleigh-Ritz algorithm.
Authors: Kafash, Behzad1 (AUTHOR) bkafash@ardakan.ac.ir
Source: Applied Numerical Mathematics. Feb2026, Vol. 220, p104-122. 19p.
Subjects: Rayleigh-Ritz method, Constrained optimization, Empirical research, Optimal control theory, Time delay systems, Approximation algorithms, Chebyshev polynomials, Mathematical optimization
Abstract: • A modified Rayleigh-Ritz method based on shifted Chebyshev polynomials is proposed for optimal control problems governed by time delay systems, with or without constraints on the control and state variables. • The method transforms the constrained time-delay optimal control problem into a constrained optimization problem and guarantees convergence analytically. • The efficiency and robustness of the proposed scheme are demonstrated through multiple case studies, including the single-input/single-output system with control and final state constraint and the harmonic oscillator under various constraint scenarios. [Display omitted] In this study, a modified Rayleigh-Ritz method is presented for the solution of optimal control problems governed by time-delayed dynamical systems, considering both constrained and unconstrained control and state variables. In this approach, the control or state variables are approximated using shifted Chebyshev polynomials with unknown coefficients. The proposed modified Rayleigh-Ritz method transforms the constrained optimal control problems governed by time-delayed dynamical systems into an optimization problem with constraints. Furthermore, a computational algorithm is developed for implementing the proposed method, and its convergence is proven analytically. To evaluate the efficiency and accuracy of the proposed algorithm, several numerical examples are presented. These include the single-input/single-output system as a case study with control and final state constraints, and an optimal control problem of the harmonic oscillator under different scenarios, which involve constraints on state and control variables. [ABSTRACT FROM AUTHOR]
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Database: Engineering Source
Description
Abstract:• A modified Rayleigh-Ritz method based on shifted Chebyshev polynomials is proposed for optimal control problems governed by time delay systems, with or without constraints on the control and state variables. • The method transforms the constrained time-delay optimal control problem into a constrained optimization problem and guarantees convergence analytically. • The efficiency and robustness of the proposed scheme are demonstrated through multiple case studies, including the single-input/single-output system with control and final state constraint and the harmonic oscillator under various constraint scenarios. [Display omitted] In this study, a modified Rayleigh-Ritz method is presented for the solution of optimal control problems governed by time-delayed dynamical systems, considering both constrained and unconstrained control and state variables. In this approach, the control or state variables are approximated using shifted Chebyshev polynomials with unknown coefficients. The proposed modified Rayleigh-Ritz method transforms the constrained optimal control problems governed by time-delayed dynamical systems into an optimization problem with constraints. Furthermore, a computational algorithm is developed for implementing the proposed method, and its convergence is proven analytically. To evaluate the efficiency and accuracy of the proposed algorithm, several numerical examples are presented. These include the single-input/single-output system as a case study with control and final state constraints, and an optimal control problem of the harmonic oscillator under different scenarios, which involve constraints on state and control variables. [ABSTRACT FROM AUTHOR]
ISSN:01689274
DOI:10.1016/j.apnum.2025.10.004