New criterion for asymptotic stability of time-varying dynamical systems

Ghrissi, Taoufik; Hammami, Mohamed; Hammi, Mekki; Mabrouk, Mohamed

doi:10.14736/kyb-2017-2-0331

Cited by 2 publications

(3 citation statements)

References 19 publications

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“…Proof. Let R(t, t 0 ) be the state transition matrix of the system (2). It follows that the solution of (1) can be written as Multiply both sides by e (t−t 0 ) and denote u(t) = ||x(t)|| e (t−t 0 ) , yields…”

Section: Resultsmentioning

confidence: 99%

“…The stability of nonautonomous perturbed dynamical systems has attracted the attention of many researchers and has produced several important results. [1][2][3][4][5] It turns out that time-varying differential equations appear as a natural description of observed evolution phenomena of various real-world problems where the study of asymptotic stability is more interesting than stability. Indeed, the study of asymptotic stability of dynamical systems is one of the most important research area in system design.…”

Section: Introductionmentioning

confidence: 99%

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New results on the uniform exponential stability of nonautonomous perturbed dynamical systems

Benjemaa

Gouadri

Hammami

2021

Intl J Robust & Nonlinear

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In this article, we investigate the asymptotic behaviors of the solutions of nonlinear dynamic systems nearby an equilibrium point, when the nominal parts are subject to non-necessarily small perturbations. We show that, under some estimates on the perturbation term, the equilibrium point remains (globally) uniformly exponentially stable. The obtained stability results can easily be applied in practice since they are based on the Gronwall-Bellman inequalities rather than the classical Lyapunov methods that require the knowledge of a Lyapunov function. Several numerical examples, as well as an application to control and mechanical systems, are given in illustration.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

New results on the uniform exponential stability of nonautonomous perturbed dynamical systems

Benjemaa

Gouadri

Hammami

2021

Intl J Robust & Nonlinear

Self Cite

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show abstract

“…Global stability refers to stability for all initial conditions, not just those in a neighborhood of the equilibrium. A Lyapunov function that is valid globally (i.e., for all 𝑥 𝜖 ℝ 𝑛 ) can establish global stability [Zhang et al, 1999;Yan & Baotong, 2006;Ghrissi et al, 2017;Liu, 2006;Carnevale et al, 2007;Vernigora, 2006]. The above-mentioned terms are visualized in Fig.…”

Section: Theoretical Backgroundmentioning

confidence: 99%

Efficient Computation of Lyapunov Functions Using Deep Neural Networks for the Assessment of Stability in Controller Design

Uyulan

2023

Preprint

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This paper presents a deep neural network (DNN) based method to estimate approximate Lyapunov functions and their orbital derivatives, which are key to the stability of the system in control theory. Our approach addresses the challenge of the curse of dimensionality in control and optimization problems, demonstrating that the computational effort required grows only polynomically with the state dimension. This is a significant improvement over traditional methods. We emphasize that the calculated functions are approximations of Lyapunov functions and not exact representations. This distinction is important, as validating these approximations in high-dimensional environments is challenging and opens new avenues for future research. Our approach diverges from traditional grid-based approaches and moves away from relying on small-gain theorems and accurate subsystem knowledge. This flexibility is proven in deriving Lyapunov functions for the development of stabilizing feedback rules in nonlinear systems. A crucial aspect of our approach is the use of ReLU activation features in neural networks, which is steady with modern deep getting-to-know traits. We also explore the feasibility of using DNNs to estimate fairly constructive Lyapunov functions, despite the demanding situations posed through uncertainty. Our set of rules' outcomes aren't unique, highlighting the want to set up criteria for figuring out especially useful Lyapunov features. The paper culminates with the aid of emphasizing the capacity of DNNs to approximate compositional Lyapunov capabilities, especially underneath small-gain conditions, to mitigate the curse of dimensionality. Our contributions are manifold, including scalability in dealing with systems of various dimensionality, flexibility in accommodating both low and excessive-dimensional structures, and performance in computing Lyapunov features through deep learning strategies. However, challenges continue to be in the approximation accuracy and the verification of Lyapunov functions in better dimensions.

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New criterion for asymptotic stability of time-varying dynamical systems

Cited by 2 publications

References 19 publications

New results on the uniform exponential stability of nonautonomous perturbed dynamical systems

New results on the uniform exponential stability of nonautonomous perturbed dynamical systems

Efficient Computation of Lyapunov Functions Using Deep Neural Networks for the Assessment of Stability in Controller Design

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