Exponential stability analysis for nonlinear time-varying perturbed systems on time scales

Cheng-Xiu Qiang; Jian-Ping Sun; Ya-Hong Zhao; Cheng-Xiu Qiang; Jian-Ping Sun; Ya-Hong Zhao

doi:10.3934/math.2023564

AIMS Mathematics

2023, Volume 8, Issue 5: 11131-11150. doi: 10.3934/math.2023564

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Exponential stability analysis for nonlinear time-varying perturbed systems on time scales

1.
College of Electrical and Information Engineering, Lanzhou University of Technology, Lanzhou 730050, China
2.
Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou 730050, China

Received: 27 December 2022 Revised: 19 February 2023 Accepted: 27 February 2023 Published: 09 March 2023
MSC : 93D05, 93D23

This paper is concerned with the stability of nonlinear time-varying perturbed system on time scales under the assumption that the corresponding linear time-varying nominal system is uniformly exponentially stable. Some less conservative sufficient conditions for uniform exponential stability and uniform practical exponential stability are proposed by imposing different assumptions on the perturbation term. Compared with the traditional exponential stability results of perturbed systems, the time derivatives of related Lyapunov functions in this paper are not required to be negative definite for all time. The main tools employed are two Gronwall's inequalities on time scales. Some examples are also given to illustrate the effectiveness of the theoretical results.
- time scale,
- nonlinear time-varying perturbed system,
- exponential stability,
- Lyapunov function
Citation: Cheng-Xiu Qiang, Jian-Ping Sun, Ya-Hong Zhao. Exponential stability analysis for nonlinear time-varying perturbed systems on time scales[J]. AIMS Mathematics, 2023, 8(5): 11131-11150. doi: 10.3934/math.2023564

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Abstract

This paper is concerned with the stability of nonlinear time-varying perturbed system on time scales under the assumption that the corresponding linear time-varying nominal system is uniformly exponentially stable. Some less conservative sufficient conditions for uniform exponential stability and uniform practical exponential stability are proposed by imposing different assumptions on the perturbation term. Compared with the traditional exponential stability results of perturbed systems, the time derivatives of related Lyapunov functions in this paper are not required to be negative definite for all time. The main tools employed are two Gronwall's inequalities on time scales. Some examples are also given to illustrate the effectiveness of the theoretical results.

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