Abstract
Lyapunov stability theory for smooth nonlinear autonomous dynamical systems is presented in terms of Geometric Algebra. The system is described by a smooth nonlinear state vector differential equation, driven by a vector field in \(\mathbb {R}^n\). The level sets of the scalar Lyapunov function candidate are assumed to be compact smooth vector manifolds in \(\mathbb {R}^n\). The level sets induce an associated global foliation of the state space. On any leaf of this foliation, a geometric subalgebra is naturally attached to the corresponding tangent vector space of the smooth vector manifold. The pseudoscalar (field) of this subalgebra completely characterizes the tangent space. Asymptotic stability of the system equilibria is described in terms of equilibria of, easily computable, rejection vector fields with respect to the pseudoscalar field. Nonexistence of invariant sets of the Lyapunov function directional derivative, along the defining vector field, are also tested using a simple tangency condition. Several illustrative examples are presented.
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Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study. The parameters used in the simulations presented are included in this article. Nevertheless, the current simulations are available from the authors on request.
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Acknowledgements
The autors are grateful to Cinvestav-IPN for its continuous support of this research. B.C. Gómez-León and M.A. Aguilar-Orduña are supported by Conacty-México, via, respectively, scholarship contract numbers: 1039577 and 702805.
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Communicated by Uwe Kaehler.
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This work was supported by “Consejo Nacional de Ciencia y Tecnología” (CONACYT-México) and “Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional” (CINVESTAV-IPN)
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Sira-Ramírez, H., Gómez-León, B.C. & Aguilar-Orduña, M.A. Lyapunov Stability: A Geometric Algebra Approach. Adv. Appl. Clifford Algebras 32, 26 (2022). https://doi.org/10.1007/s00006-022-01210-6
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DOI: https://doi.org/10.1007/s00006-022-01210-6