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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
##### Keywords:
linear delay systems; bounded variation; eigenvalues
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##### References:
 [1] Hale, J.K, Theory of functional differential equations, (1977), Springer-Verlag New York · Zbl 0425.34048 [2] Bellman, R; Cooke, K.L, Differential-difference equations, (1963), Academic Press New York · Zbl 0118.08201 [3] Infante, E.F; Castelan, W.B, A Liapunov functional for a matrix difference-differential equation, J. differential equations, 29, 439-451, (1978) · Zbl 0354.34049 [4] Doetsch, G, Introduction to theory and application of the Laplace transformation, (1974), Springer-Verlag New York/Heidelberg/Berlin · Zbl 0278.44001 [5] Serge, L, Complex analysis, (1977), Springer-Verlag New York
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