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A sufficient condition for the existence of a Lyapunov function of the form *V*(*x*) = *x*^{T} ^{x}*P*, *P* = *P*^{T} > 0, *P* ∈ R^{n×n}, for the stable linear time invariant systems *ẋ* = _{i}*A x*, _{i}*A* ∈ R^{n×n}, _{i}*A* ∈ *A* ≜ {*A*_{1}, …, _{m}*A*}, is that the matrices _{i}*A* are Hurwitz, and that a non‐singular matrix *T* exists, such that _{i}*T A T*^{−1}, *i* ∈ {1, …, *m*}, is upper triangular [Mori, Y., Mori, T., & Kuroe, Y., *Proceedings of Electronic Information and Systems Conference* (1996); *Proceedings of 36th Conference on Decision and Control*, (1997); Liberzon, D., Hespanha, J.P., & Morse, S., *Technical Report, Laboratory for Control Science and Engineering, Yale University*, (1998); Shorten, R. & Narendra, K., *Proceedings of Conference on Decision and Control*, (1998)]. The existence of such a function, referred to as a *common quadratic Lyapunov function* (CQLF), is sufficient to guarantee the exponential stability of the switching system *ẋ* = *A*(*t*)x, *A*(*t*) ∈ *A*. In this paper we investigate the stability properties of a related class of switching systems. We consider sets of matrices *A*, where no single matrix *T* exists that simultaneously transforms each _{i}*A* ∈ *A* to upper triangular form, but where a set of non‐singular matrices _{i j}*T* exist such that the matrices {_{i j}_{i}*T A* _{i j}*T*^{−1}, _{i j}_{j}*T A* _{i j}*T*^{−1}}, *i*, *j* ∈ {1, …, *m*}, are upper triangular. We show that, for a special class of such systems, the origin of the switching system *ẋ* = *A*(*t*)*x*, *A*(*t*) ∈ *A*, is globally attractive. A novel technique is developed to derive this result, and the applicability of this technique to more general systems is discussed towards the end of the paper.

*Keywords: *
Stability; switching‐; systems; hybrid‐; systems; Lyapunov

*Journal Article.*
*0 words.*

*Subjects: *
Mathematics

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