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Generalization of Liapunov’s theorem in a linear delay system. (English) Zbl 0705.34084
The author extends the work of A. M. Lyapunov to linear delay systems \((*)\quad x'(t)=L(x_ t)=\int^{0}_{-r}d\eta (\theta)x_ t(\theta),\) where L is a linear operator on \(C([-r,0],R^ n)\), that is, \(\eta\) is an \(n\times n\) matrix of bounded variation on [-r,0]. He obtains the following result: Theorem. If \(\lambda_ 1+\lambda_ 2\neq 0\) for any pair of eigenvalues \(\lambda_ 1\) and \(\lambda_ 2\) of L, then (i) For any given \(n\times n\) symmetric matrix W, there is a functional \(V: C([- r,0],R^ n)\to R\) such that \(\dot V_{(*)}(\phi)=-\phi '(0)W\phi (0),\phi \in C([-r,0),R^ n)=C\). (ii) If, in addition, all eigenvalues of L have negative real parts and W is positive definite, then for any given \(\alpha >0\), there are monotone increasing functions \(u_{\alpha}\), \(v\in C(R^+,R^+)\), \(u_{\alpha}(0)=v(0)=0\) such that \(u_{\alpha}(| \phi (0)|)\leq V(\phi)\leq v(| \phi |)\), \(\phi \in C_{\alpha}=\{\Psi \in C:| \Psi | \leq \alpha \}\).
Reviewer: N.Parhi

MSC:
34K20 Stability theory of functional-differential equations
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References:
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