Abstract
In this chapter we introduce the class of control systems, tautological control systems, that we propose as being useful mathematical models for the investigation of geometric system structure. As promised in our introduction in Sect. 1.2, this class of systems naturally handles a variety of regularity classes; we work with finitely differentiable, Lipschitz, smooth, and real analytic classes simultaneously with comparative ease. Also as indicated in Sect. 1.2, the framework makes essential use of sheaf theory in its formulation. We shall see in Sect. 5.6 that the natural morphisms for tautological control systems ensure feedback-invariance of the theory.
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Notes
- 1.
This relies on the fact that Oka’s Theorem, in the version of “the sheaf of sections of a vector bundle is coherent”, holds in the real analytic case. It does, and the proof is the same as for the holomorphic case [8, Theorem 3.19] since the essential ingredient is the Weierstrass Preparation Theorem, which holds in the real analytic case [17, Theorem 6.1.3].
- 2.
Recall that this means that \(\mathcal {{C}}\) is the continuous image of a complete, separable, metric space. We refer to [3, §6.6–6.8] for an outline of the theory of Suslin spaces.
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Lewis, A.D. (2014). Tautological Control Systems: Definitions and Fundamental Properties. In: Tautological Control Systems. SpringerBriefs in Electrical and Computer Engineering(). Springer, Cham. https://doi.org/10.1007/978-3-319-08638-5_5
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