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Attracting and natural invariant varieties for polynomial vector fields and control systems. (English) Zbl 1433.93040

The authors 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 they consider the construction of polynomial ordinary differential equations \(\dot{x} = f(x)\) 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 they 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 \(\dot{x} = f(x) + g(x)u\) for any choice of the drift vector \(f\). They use basic tools from commutative algebra and algebraic geometry in order to characterize NCIV’s, and they 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.

MSC:

93B52 Feedback control
34C45 Invariant manifolds for ordinary differential equations
93C10 Nonlinear systems in control theory
93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems)

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References:

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