Exponential stabilization of quasi-one-sided Lipschitz systems with time delay

Omar Kahouli; Lilia El Amraoui; Mohamed Ayari; Hamdi Gassara; Ahmed El Hajjaji; Omar Kahouli; Lilia El Amraoui; Mohamed Ayari; Hamdi Gassara; Ahmed El Hajjaji

doi:10.3934/math.20251173

AIMS Mathematics

2025, Volume 10, Issue 11: 26680-26696. doi: 10.3934/math.20251173

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Exponential stabilization of quasi-one-sided Lipschitz systems with time delay

1.
Department of Electronics Engineering, Applied College, University of Ha'il, Ha'il 2440, Saudi Arabia
2.
Department of Electrical Engineering, College of Engineering, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
3.
Department of Information Technology, Faculty of Computing and Information Technology, Northern Border University, Arar 91431, Saudi Arabia
4.
Laboratory of Sciences and Technology of Automatic Control and Computer Engineering, National School of Engineering of Sfax, University of Sfax, P.O. Box 1173, Sfax 3038, Tunisia
5.
Modeling, Information, and Systems Laboratory, University of Picardie Jules Verne, Amiens 80000, France

Received: 16 September 2025 Revised: 17 October 2025 Accepted: 05 November 2025 Published: 18 November 2025
MSC : 93D23, 93C10, 93C43

This paper aims to design Observer-Based (OB) controllers that ensure the exponential stability of a class of nonlinear time-delay systems. The nonlinear part of the system satisfies a weak Quasi-One-Sided Lipschitz (QOSL) condition characterized by the matrices $ (\mathcal{L}_1, \mathcal{M}_1, \mathcal{N}_1) $, as well as a QOSL condition characterized by the matrices $ (\mathcal{L}_2, \mathcal{M}_2, \mathcal{N}_2) $. First, we derive a sufficient condition formulated as a Linear Matrix Inequality (LMI) via a Lyapunov–Krasovskii (LK) functional. The main advantage of this design is that the controller and observer gains are computed in a single step. However, its main drawback is that the matrices $ \mathcal{L}_1 $, $ \mathcal{M}_1 $, and $ \mathcal{N}_1 $ are fixed rather than treated as decision variables. To overcome this limitation, we propose an improved design in which the matrices $ \mathcal{L}_1, \; \mathcal{M}_1 $, and $ \mathcal{N}_1 $ are treated as decision-variable matrices with a fixed structure. By using an appropriate decoupling technique, this approach provides greater flexibility in the selection of matrices and reduces conservatism. The efficacy of the developed OB controllers is demonstrated via a suitable numerical example.
- exponential stability,
- observer-based controller,
- time-delay systems,
- quasi-one-sided Lipschitz condition,
- decoupling technique
Citation: Omar Kahouli, Lilia El Amraoui, Mohamed Ayari, Hamdi Gassara, Ahmed El Hajjaji. Exponential stabilization of quasi-one-sided Lipschitz systems with time delay[J]. AIMS Mathematics, 2025, 10(11): 26680-26696. doi: 10.3934/math.20251173

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Abstract

This paper aims to design Observer-Based (OB) controllers that ensure the exponential stability of a class of nonlinear time-delay systems. The nonlinear part of the system satisfies a weak Quasi-One-Sided Lipschitz (QOSL) condition characterized by the matrices $ (\mathcal{L}_1, \mathcal{M}_1, \mathcal{N}_1) $, as well as a QOSL condition characterized by the matrices $ (\mathcal{L}_2, \mathcal{M}_2, \mathcal{N}_2) $. First, we derive a sufficient condition formulated as a Linear Matrix Inequality (LMI) via a Lyapunov–Krasovskii (LK) functional. The main advantage of this design is that the controller and observer gains are computed in a single step. However, its main drawback is that the matrices $ \mathcal{L}_1 $, $ \mathcal{M}_1 $, and $ \mathcal{N}_1 $ are fixed rather than treated as decision variables. To overcome this limitation, we propose an improved design in which the matrices $ \mathcal{L}_1, \; \mathcal{M}_1 $, and $ \mathcal{N}_1 $ are treated as decision-variable matrices with a fixed structure. By using an appropriate decoupling technique, this approach provides greater flexibility in the selection of matrices and reduces conservatism. The efficacy of the developed OB controllers is demonstrated via a suitable numerical example.

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