Abstract :
The equations of the title for the scalar state x: [− T, + ∞) → R are ẋ(t) − ax(t) + Bx(t − T) = 0, a ≔ [a, a], B ≔ [B, B], T ≔ [T, T], (1) where a, B, T are non-empty real intervals. Such equations occur upon linearization when investigating the local asymptotical stability of equilibria in non-linear dynamical systems of the form ẏ(t) = F[y(t), y(t − T)], T> 0, (2) with uncertain dynamics F: R2 → R. In control theory, (1) is discussed in context of adaptive and robust control. Given a, B, T, we wish to determine whether or not the equilibrium x(t)≡ x ≔ 0 of (1) is asymptotically stable. In terms of the characteristic functions, H(z), for (1), H(z) ≔ z − a + Be−zT, z ∈ C, (3)we have to decide whether or not all zeros of H(z) lie in Re(z) < 0 for given a, B, T. We aim at an explicit result and focus on the set of all T ∈ R+ for which x* is asymptotically stable. In the case when the intervals shrink to points, a ≔ [a, a], B ≔ [B, B], T ≔ [T, T], there is an explicitly known stability boundary, T(a, B) ≥ 0, such that x* is asymptotically stable if and only if T lies in an open initial interval, 0 ≤ T ≤ T(a,B), (4)which may be void, T(a, B) ≔ 0, of finite length, T(a, B) ≤ ∞, or of infinite length, T(a, B) ≔ ∞. The explicit knowledge of T(a, B) enables us to generalize (4) to the case of intervals a, B of positive length, T ⊂ (0, T(a,B)) (5)with explicit T(a, B). The result is discussed in the light of the stability criterion of Kharitonov [Differentsial′nye Uravneiva 14, No. 11 (1978), 2086-2088] for polynomials.
