Operator-Norm Transfer and Cohomological Rigidity for Quaternionic Quasi-Lie Structures with Application to Sliding Mode $β$-Exponential Stability
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| Formato: | Preprint |
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2026
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| _version_ | 1866913168150757376 |
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| author | Athmouni, Nassim Brahmia, Nejib Fajraoui, Tarek Mabrouk, Fehmi |
| author_facet | Athmouni, Nassim Brahmia, Nejib Fajraoui, Tarek Mabrouk, Fehmi |
| contents | We develop operator-theoretic and cohomological tools for quaternionic quasi-Lie structures, with sliding mode control as a motivating application. Three main results are established. First, an exact operator-norm transfer under the quaternionic anti-isomorphism $a \mapsto \bar{a}$, which enables quantitative bounds from~\cite{Athmouni2026} to transfer between left- and right-module conventions with no constant factor. Second, a transcription of the cohomological rigidity result of~\cite{Athmouni2026} into a form usable here: in the homogeneous case, under a local cohomological non-obstruction hypothesis, an explicit bilinear correction $Ω$ produces a bracket satisfying the Jacobi identity exactly on a ball of admissible radius, with all quantitative constants expressed through $C_{2}$ and the admissible radius. Third, the projected Jacobi defect is shown to satisfy a generalized one-sided Lipschitz condition with computable constants, obtained via a uniform-selection argument handling state-dependence of the measurable selection. As an application, we develop a robust control framework with a cohomological matching condition replacing pointwise verification: an integral sliding surface yields $β$-exponential stability via an iterative linear matrix inequality (LMI) scheme. The work is purely analytical; closed-loop numerical simulations for multidimensional systems are deferred to a companion paper. The framework is restricted to the homogeneous quasi-Lie case; the Sobolev extension is conjectural, and algorithm termination is established conditionally on sufficient continuity assumptions on the LMI solution map. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_28385 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Operator-Norm Transfer and Cohomological Rigidity for Quaternionic Quasi-Lie Structures with Application to Sliding Mode $β$-Exponential Stability Athmouni, Nassim Brahmia, Nejib Fajraoui, Tarek Mabrouk, Fehmi Optimization and Control Functional Analysis Rings and Algebras 47S10, 17B56, 93B12, 93C10, 93D09, 93D23, 15B33, 17A30, 46S10 Secondary 93B12, 93C10, 93D09, 93D23, 15B33, 17A30, 46S10 We develop operator-theoretic and cohomological tools for quaternionic quasi-Lie structures, with sliding mode control as a motivating application. Three main results are established. First, an exact operator-norm transfer under the quaternionic anti-isomorphism $a \mapsto \bar{a}$, which enables quantitative bounds from~\cite{Athmouni2026} to transfer between left- and right-module conventions with no constant factor. Second, a transcription of the cohomological rigidity result of~\cite{Athmouni2026} into a form usable here: in the homogeneous case, under a local cohomological non-obstruction hypothesis, an explicit bilinear correction $Ω$ produces a bracket satisfying the Jacobi identity exactly on a ball of admissible radius, with all quantitative constants expressed through $C_{2}$ and the admissible radius. Third, the projected Jacobi defect is shown to satisfy a generalized one-sided Lipschitz condition with computable constants, obtained via a uniform-selection argument handling state-dependence of the measurable selection. As an application, we develop a robust control framework with a cohomological matching condition replacing pointwise verification: an integral sliding surface yields $β$-exponential stability via an iterative linear matrix inequality (LMI) scheme. The work is purely analytical; closed-loop numerical simulations for multidimensional systems are deferred to a companion paper. The framework is restricted to the homogeneous quasi-Lie case; the Sobolev extension is conjectural, and algorithm termination is established conditionally on sufficient continuity assumptions on the LMI solution map. |
| title | Operator-Norm Transfer and Cohomological Rigidity for Quaternionic Quasi-Lie Structures with Application to Sliding Mode $β$-Exponential Stability |
| topic | Optimization and Control Functional Analysis Rings and Algebras 47S10, 17B56, 93B12, 93C10, 93D09, 93D23, 15B33, 17A30, 46S10 Secondary 93B12, 93C10, 93D09, 93D23, 15B33, 17A30, 46S10 |
| url | https://arxiv.org/abs/2605.28385 |