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

Qualitative Theory of Dynamical Systems, vol. 19

Abstract

Abstract 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 $$\dot{x}=f(x)$$ 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 $${\dot{x}}=f(x)+g(x)u$$ 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.

Authors 4

1. Niclas Kruff Aachen Lehrstuhl A für Mathematik

   RWTH Aachen University

   Affiliation as printed

   Lehrstuhl A für Mathematik, RWTH Aachen, 52056, Aachen, Germany

2. Christian Schilli Aachen Lehrstuhl D für Mathematik

   RWTH Aachen University

   Affiliation as printed

   Lehrstuhl D für Mathematik, RWTH Aachen, 52056, Aachen, Germany

3. Sebastian Walcher corresponding Aachen Lehrstuhl A für Mathematik

   RWTH Aachen University

   Affiliation as printed

   Lehrstuhl A für Mathematik, RWTH Aachen, 52056, Aachen, Germany

4. Eva Zerz Aachen Lehrstuhl D für Mathematik

   RWTH Aachen University

   Affiliation as printed

   Lehrstuhl D für Mathematik, RWTH Aachen, 52056, Aachen, Germany