Time-scaling symmetry and Zeno solutions
type: Item
instance of: scholarly article
title: Time-scaling symmetry and Zeno solutions
zbMATH Open document ID: 1173.37026
published in: Automatica
publication date: 2009-06-10T00:00:00Z
review text: The article deals with the analysis of dynamical systems, described with the differential equation \[ \dot x(t) = f(x(t)); \] the function \(f(\cdot)\) is not assumed to be continuous. However, it is assumed that this system exhibits time-scaling symmetry with respect to a given spatial transformation group \({\mathcal F} = \{F_\alpha: {\mathbb R}^n \to {\mathbb R}^n, \;\alpha \in (0,\infty)\}\) if, for any solution \(x(t)\) and any \(\alpha > 0\) the function \(T_\alpha x(t) = F_\alpha(x(\alpha t))\) is also a solution (on the corresponding time-interval). As examples of such systems the well known Fuller system \[ \dot x_1(t) = x_2(t), \quad \dot x_2(t) = - \text{sgn} \, (x_1(t) + \frac12 m x_2^2(t) \;\text{sgn} \, x_2(t)) \] (with the group \(F_\alpha = \text{diag} \, (\alpha^{-2},\alpha^{-1})\)), Filippov system \[ \dot x_1(t) = \text{sgn} \, x_1(t) - 2 \;\text{sgn} \, x_2(t), \quad \dot x_2(t) = 2 \;\text{sgn} \, x_1(t) + \text{sgn} \, x_2(t) \] (with the group \(F_\alpha = \text{diag} \, (\alpha^{-1},\alpha^{-1})\)) and Pogromsky -- Heemels -- Nijmeijer system \[ \dot x_1(t) = x_2(t), \quad \dot x_2(t) = x_3(t), \quad \dot x_3(t) = - \text{sgn} \, x_1(t) \] are considered. The first result in the article is the statement that, for an invariant set \(S\) of the system, the spatial transformation group restricted to \(S\) commutes with the Poincaré return map; the latter is defined by the formula \(Rx_0 = x(\tau(x_0),x_0)\), \(\tau(x_0)\) is the first return time of \(x_0\). The second result is concerned with the system that is described with the equation \[ \dot x(t) = f(x) = Ax(t) - b \;\text{sgn} \, (c^Tx(t)); \] under this it is assumed that there exists a matrix \(K\) such that \(AK - KA = A\), \(Kb = - b\), \(c^TK = \lambda c^T\) with \(\lambda < 0\) and \(c^TA^{-\lambda-1}b = 1\). If there exists \(\alpha > 1\) such that the matrix \(I + e^{-A} \alpha^K\) is invertible, \[ h(\alpha) \doteq c^T(I + e^{-a}\alpha^K)^{-1} \int_0^1 e^{-As} \, ds \, b = 0, \] …
Mathematics Subject Classification ID: 37C80
zbMATH DE Number: 5563967
zbMATH Keywords: time-scaling symmetry
zbMATH Keywords: Zeno solutions
zbMATH Keywords: relay sistems
zbMATH Keywords: Poincaré return map
zbMATH Keywords: hybrid systems
author: Johannes M. Schumacher
reviewed by: Peter P. Zabreĭko
MaRDI profile type: Publication
full work available at URL: j.automatica.2008.12.008
OpenAlex ID: W2059033614
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DOI: 10.1016/J.AUTOMATICA.2008.12.008
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description: scientific article; zbMATH DE number 5563967