Shifted Legendre polynomials-based single and double integral inequalities with arbitrary approximation order: Application to stability of linear systems with time-varying delays

Deren Gong; Xiaoliang Wang; Peng Dong; Shufan Wu; Xiaodan Zhu; Deren Gong; Xiaoliang Wang; Peng Dong; Shufan Wu; Xiaodan Zhu

doi:10.3934/math.2020279

AIMS Mathematics

2020, Volume 5, Issue 5: 4371-4398. doi: 10.3934/math.2020279

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Research article

Shifted Legendre polynomials-based single and double integral inequalities with arbitrary approximation order: Application to stability of linear systems with time-varying delays

1.
School of Aeronautics and Astronautics, Shanghai Jiaotong University, Dongchuan Rd. No. 800, Shanghai 200240, China
2.
Chengdu Aircraft Design and research Institute (CADI), Riyue Avenue No.1610, Chengdu, China

Received: 07 December 2019 Accepted: 06 May 2020 Published: 11 May 2020
MSC : 93D99

This paper proposes novel single and double integral inequalities with arbitrary approximation order by employing shifted Legendre polynomials and Cholesky decomposition, and these inequalities could significantly reduce the conservativeness in stability analysis of linear systems with interval time-varying delays. The coefficients of the proposed single and double integral inequalities are determined by using the weighted least-squares method. Also former well-known integral inequities, such as Jensen inequality, Wirtinger-based inequality, auxiliary function-based integral inequalities, are all included in the proposed integral inequalities as special cases with lower-order approximation. Stability criterions with less conservatism are then developed for both constant and time-varying delay systems. Several numerical examples are given to demonstrate the effectiveness and benefit of the proposed method.
- shifted Legendre polynomials,
- Cholesky decomposition,
- arbitrary approximation order,
- time-delayed systems,
- stability analysis
Citation: Deren Gong, Xiaoliang Wang, Peng Dong, Shufan Wu, Xiaodan Zhu. Shifted Legendre polynomials-based single and double integral inequalities with arbitrary approximation order: Application to stability of linear systems with time-varying delays[J]. AIMS Mathematics, 2020, 5(5): 4371-4398. doi: 10.3934/math.2020279

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Abstract

This paper proposes novel single and double integral inequalities with arbitrary approximation order by employing shifted Legendre polynomials and Cholesky decomposition, and these inequalities could significantly reduce the conservativeness in stability analysis of linear systems with interval time-varying delays. The coefficients of the proposed single and double integral inequalities are determined by using the weighted least-squares method. Also former well-known integral inequities, such as Jensen inequality, Wirtinger-based inequality, auxiliary function-based integral inequalities, are all included in the proposed integral inequalities as special cases with lower-order approximation. Stability criterions with less conservatism are then developed for both constant and time-varying delay systems. Several numerical examples are given to demonstrate the effectiveness and benefit of the proposed method.

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