Attracting and Natural Invariant Varieties for Polynomial Vector Fields and Control Systems

• Kruff Niclas ^[1] ; Schilli Christian ^[1] ; Walcher, Sebastian ^[1] ; Zerz Eva ^[1]
  1. [1] Rheinisch-Westfälische Technische Hochschule Aachen University

     Rheinisch-Westfälische Technische Hochschule Aachen University

     Städteregion Aachen, Alemania

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 19, Nº 1, 2020
• Idioma: inglés
• DOI: 10.1007/s12346-020-00365-6
• Enlaces
  • Texto completo
• Resumen

  • We discuss real and complex polynomial vector fields and polynomially nonlinear, input-affine control systems, with a focus on invariant algebraic varieties. For a given real variety we consider the construction of polynomial ordinary differential equations x˙=f(x) such that the variety is invariant and locally attracting, and show that such a construction is possible for any compact connected component of a smooth variety satisfying a weak additional condition. Moreover we introduce and study natural controlled invariant varieties (NCIV) with respect to a given input matrix g, i.e. varieties which are controlled invariant sets of x˙=f(x)+g(x)u for any choice of the drift vector f. We use basic tools from commutative algebra and algebraic geometry in order to characterize NCIV’s, and we present a constructive method to decide whether a variety is a NCIV with respect to an input matrix. The results and the algorithmic approach are illustrated by examples.

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