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Absolute and input-to-state stabilities of nonautonomous systems with causal mappings. (English) Zbl 1185.34103

Summary: We consider systems governed by the scalar equation
\[ \sum^n_{k=0}a_k(t)x^{(n-k)}(t)=[Fx](t)\quad (t\geq 0), \]
where \(a_0\equiv 1\); \(a_k(t)\) \((k=1,\dots,n)\) are positive continuous functions and \(F\) is a causal mapping. We also consider the case when \(F\) depends on the input. Such equations include differential, integrodifferential and other traditional equations. It is assumed that all the roots \(\tau_k(t)\) \((k=1,\dots,n)\) of the polynomial \(z^n+a_1(t)z^{n-1}+\cdots+a_n(t)\) are real and negative for all \(t\geq 0\). Exact explicit conditions for the absolute and input-to-state stabilities of the considered systems are established.

MSC:

34K20 Stability theory of functional-differential equations
34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
93D25 Input-output approaches in control theory
34D20 Stability of solutions to ordinary differential equations
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