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| Autor principal: | |
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| Formato: | Recurso digital |
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Zenodo
2025
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| Acesso em linha: | https://doi.org/10.5281/zenodo.17686740 |
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Sumário:
- Singular matrix pencils, represented as $A - lambda B$, where $A$ and $B$ are rectangular or square but generically rank-deficient matrices, are fundamental structures in diverse fields such as control theory, descriptor systems, and constrained dynamics. While their algebraic properties are extensively characterized by the Kronecker canonical form and associated invariants like finite and infinite eigenvalues, and minimal indices, these algebraic measures often provide a localized and discrete perspective. This paper introduces and explores the concept of topological invariants for singular matrix pencils, aiming to capture their global qualitative features and robustness under continuous perturbations. We develop a theoretical framework leveraging concepts from algebraic topology, specifically by associating vector bundles with the null spaces and range spaces of the pencil over the Riemann sphere. We investigate how characteristic classes, such as Chern classes, can serve as novel topological invariants that describe the continuous deformation properties and structural stability of these pencils. The methodology involves analyzing the geometric variations of these subspaces as functions of the spectral parameter $lambda$. Our results demonstrate that these topological invariants offer a complementary and deeper understanding of the intrinsic properties of singular systems, providing insights into their robustness and qualitative behavior that transcend purely algebraic classifications. This work opens new avenues for the analysis and design of robust control strategies for descriptor systems and other applications involving singular pencils.