Absolute and input-to-state stabilities of nonautonomous systems with causal mappings

We consider systems governed by the scalar equation Σ _k=0ⁿa_k(t)x^(n-k)(t) = [F _x](t) (t ≥ 0), where a_o ≡ 1; a_k(t) (k = 1,... ,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 r_k(t) (k = l,...,n) of the polynomial z ⁿ + a₁(t)z^n-1+...+a_n(t) are real and negative for all t > 0. Exact explicit conditions for the absolute and input-to-state stabilities of the considered systems are established