# zbMATH — the first resource for mathematics

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)

##### MSC:
 93D20 Asymptotic stability in control theory 93B30 System identification 93C99 Model systems in control theory
##### Keywords:
time-varying; identification; asymptotic stability
Full Text:
##### References:
 [1] D. Aeyels: Stability of nonautonomous systems by Liapunov’s direct method. the Proceedings of the Workshop Geometry in Nonlinear Control, Banach Center Publications, June 1993, to appear. · Zbl 0877.93115 · doi:10.1016/0167-6911(94)00088-D [2] B. D. O. Anderson: Exponential Stability of linear equations arising in adaptive identification. IEEE Trans. Automat. Control 22 (1977), 84-88. · Zbl 0346.93014 · doi:10.1109/TAC.1977.1101406 [3] H. K. Khalil: Nonlinear Systems. McMillan Publishing Company, New York 1992. · Zbl 0969.34001 [4] T. Kohonen: Self-organization and associative memory. (Springer Ser. Inform. Sci. 8.) Springer-Verlag, Berlin 1988. · Zbl 0659.68100 [5] G. Kreisselmeier: Adaptive observers with exponential rate of convergence. IEEE Trans. Automat. Control 22 (1977), 1, 2-8. · Zbl 0346.93043 · doi:10.1109/TAC.1977.1101401 [6] K. S. Narendra, A. M. Annaswamy: Persistent excitation in adaptive systems. Internat. J. Control 45 (1987), 1, 127-160. · Zbl 0627.93041 · doi:10.1080/00207178708933715 [7] N. Rouche P. Habets, M. Laloy: Stability Theory by Liapunov’s Direct Method. Springer-Verlag, New York 1977. · Zbl 0364.34022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.