Stability Theorems for Delay Differential Inclusions

Liu, Kun‐Zhi; Liu, Jun; Teel, Andrew R.

doi:10.1109/tac.2015.2507782

Cited by 23 publications

(20 citation statements)

References 29 publications

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“…Proof: Stability is a direct result in [17]. Attractivity follows from the invariance principle in Theorem 1.□…”

Section: Then the Solution ( ) = 0 Is Stable Moreover If The Largesmentioning

confidence: 98%

“…On the existence and continuation of solutions of ℐ, details can be found in [8], [17]. Given a solution of ℐ, ( ) is the trajectory of in the space .…”

Section: Preliminariesmentioning

confidence: 99%

“…Given a continuous functional defined on , the upper right-hand derivative of functional for some ∈ ℝ is given, in the constructive way ( [19], [3], [17]), as…”

Section: Preliminariesmentioning

confidence: 99%

“…Delay differential inclusions can describe a wide variety of dynamic systems affected by delays and can also be used to analyze stability and robustness of delay differential equations with discontinuous right-hand sides [17]. This paper is aimed at developing the classical invariance principle [12], [10] for delay differential inclusions.…”

Section: Introductionmentioning

confidence: 99%

“…However, stability criteria are still not well developed for such a class of widely used systems. Some stability theorems for delay differential inclusions have been developed in [17] and [24] where [17] admits a more general functional. Only invariantly differentiable functional is used in [24] and the results in [24] cannot be applied to nonautonomous delay differential inclusions with only weak Lyapunov functional.…”

Section: Introductionmentioning

confidence: 99%

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Invariance principles for delay differential inclusions

Liu

Sun

Wang

et al. 2015

The 27th Chinese Control and Decision Conference (2015 CCDC)

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This paper establishes two invariance principles for delay differential inclusions. The delay differential inclusions are required to satisfy the basic assumptions: the right-hand sides are upper semicontinuous and take nonempty compact and convex values on the domains. The classical LaSalle's invariance principle for delay differential inclusions is established successfully by locally Lipschitz Lyapunov-Krasovskii functionals and several stability corollaries are developed. Besides, the concept of limit delay differential inclusions is proposed to generalize the invariance principle to time-varying delay differential inclusions. Some numerical examples are given to show the effectiveness of the proposed results.

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“…Proof: Stability is a direct result in [17]. Attractivity follows from the invariance principle in Theorem 1.□…”

Section: Then the Solution ( ) = 0 Is Stable Moreover If The Largesmentioning

confidence: 98%

“…On the existence and continuation of solutions of ℐ, details can be found in [8], [17]. Given a solution of ℐ, ( ) is the trajectory of in the space .…”

Section: Preliminariesmentioning

confidence: 99%

“…Given a continuous functional defined on , the upper right-hand derivative of functional for some ∈ ℝ is given, in the constructive way ( [19], [3], [17]), as…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Invariance principles for delay differential inclusions

Liu

Sun

Wang

et al. 2015

The 27th Chinese Control and Decision Conference (2015 CCDC)

Self Cite

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Finite‐time/fixed‐time stability and stabilization of a class of delayed neural networks system with discontinuous activation

Zhang,

Zhao,

Pan

et al. 2024

Asian Journal of Control

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This paper focuses to analyze the finite‐time stability (FNTS) and fixed‐time stability (FXTS) of a class of discontinuous neural networks (NNs) with time‐varying delays. Firstly, considering the fuzzy logics and discontinuities in the NNs, a novel FNTS/FXTS approach based functional differential inclusion (FDI) theory is proposed. Under the framework of the Filippov state solution, the generalized Lyapunov functional method developed can obtain an indefinite time derivative almost anywhere along the system's state solutions. Some novel FNTS and FXTS analysis of the trivial state solution for FDI are provided. In particular, the settling time (ST) of FNTS/FXTS is presented. For the requirements of FNTS and FXTS, two control laws are designed for the discontinuous delayed neural networks (DNNs), respectively. Finally, the correctness and the advantages of the proposed method are demonstrated through its DNNs applications.

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Lyapunov‐Krasovskii stability analysis of delayed Filippov system: Applications to neural networks with switching control

Cai

Huang

2019

Intl J Robust & Nonlinear

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Summary This paper gives the stability analysis of Filippov system with delay. Under the Filippov‐framework, we introduce a general type of Lyapunov‐Krasovskii functional (LKF) to derive the stability of delayed differential inclusions (DDIs), where the indefiniteness or positive definiteness of the derivative of LKF holds for almost everywhere along the trajectories of state solution. The proposed LKF of this paper generalizes the classic LKF whose derivative possesses negative definiteness or seminegative definiteness for everywhere. As a result, the stability, uniform stability, uniform asymptotic stability, and global exponential stability criteria of the trivial solution for DDI are established. Moreover, the developed LKF method is applied to solve the stabilization control issue of delayed neural networks, possessing discontinuous input‐output activation.

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Stability Theorems for Delay Differential Inclusions

Cited by 23 publications

References 29 publications

Invariance principles for delay differential inclusions

Invariance principles for delay differential inclusions

Finite‐time/fixed‐time stability and stabilization of a class of delayed neural networks system with discontinuous activation

Lyapunov‐Krasovskii stability analysis of delayed Filippov system: Applications to neural networks with switching control

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