Abstract
In this paper, we study the absolute equivalence between Pfaffian systems with a degree 1 independence condition and obtain structural results, particularly for systems of corank 3. We apply these results to understanding dynamic feedback linearization of control systems with 2 inputs.
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Notes
We will call such a Pfaffian system a system of corank 2.
By “generic,” we mean that this condition holds for a set of integral curves that is open and dense in the space of all integral curves with respect to the \(C^k\) topology for some k. We do not require all integral curves of \((M,I;\tau )\) to satisfy the condition, in order to make the concept useful even when there exists a negligible set of integral curves that do not admit unique liftings. For example, consider \((M,I;\tau ):= (\mathbb R^5, [\![\textrm{d}x - u \textrm{d}t, \textrm{d}y - v\textrm{d}t]\!]; \textrm{d}t)\). It has a Cartan prolongation: \((N,J;\sigma ):= (\mathbb R^5\times \mathbb R, [\![I, \textrm{d}u - \lambda \textrm{d}v]\!]; \textrm{d}t)\), to which all integral curves of \((M,I;\tau )\) admits a unique lifting except those along which v is a constant.
The reader may compare this with [10, p.61].
Here and below, we drop the independence conditions in these diagrams for clarity.
Compare with [11, Definition 3.2].
This theorem may be seen as a variant of [11, Theorem 3.6].
In fact, this prolongation is obtained by setting \(n = 3\) in Example 2.34.
Here we do not require each prolongation by differentiation to be constructed from a fixed set of coordinates.
We adopt the convention of summing over repeated indices.
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Acknowledgements
We would like to thank Prof. Robert L. Bryant for generously offering his time for discussion. This work also benefited from discussions with Prof. George R. Wilkens and Taylor J. Klotz. Additionally, we would like to thank the referee for their comments and questions, which have led to substantial improvements in our exposition.
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The first author was supported by a Collaboration Grant for Mathematicians from the Simons Foundation, and the second author was partially supported by the China Postdoctoral Science Foundation Grant 2021TQ0014.
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Clelland, J.N., Hu, Y. On absolute equivalence and linearization I. Geom Dedicata 217, 78 (2023). https://doi.org/10.1007/s10711-023-00811-0
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DOI: https://doi.org/10.1007/s10711-023-00811-0