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On the convergence of a time-variant linear differential equation arising in identification. (English) Zbl 0832.93051
Given the time-varying differential equation (arising in identification and control problems): \(\dot x= -m(t) m^T(t) x\), where \(x\) and \(m\) are \(n\)-dimensional vectors, the authors formulate the following main result. Let \(m(t)\) be regulated and bounded. A sufficient condition for asymptotic stability of the given system is that for each \(t\) there exists a \(T(t)> t\) such that \(\alpha I\leq \int^{T(t)}_t m(\tau) m^T(\tau) d\tau\leq \beta I\), with \(\beta\geq \alpha> 0\). The role of the bounds \(\alpha\) and \(\beta\) are discussed and it is pointed out that the results still hold if \(m\) is replaced by an \(n\times p\) matrix, with \(p\leq n\).
Reviewer: M.Voicu (Iaşi)

93D20 Asymptotic stability in control theory
93B30 System identification
93C99 Model systems in control theory
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