Stability of a chain of linear differential equations

The behavior of an infinite sequence of ordinary differential equations of the form:

$\text{dXn}\text{dt}\text{=}\text{∑}\text{i=?M}\text{N}\text{L}{}_{\mathrm{i}}\text{X}{}_{\mathrm{i+n}}\text{, 0 ? n, 0 < N, M < ∞}$

,

$\text{X}{}_{\mathrm{n}}\text{(0) = C}{}_{\mathrm{n}}\text{, (1) X}{}_{\mathrm{n}}\text{≡ 0, n < 0}$

, where X_n(t) is a vector valued function of R⁺, is studied in spaces of infinite sequences of vectors. In particular, sufficient conditions for asymptotic stability of this sequence of linear equations are established and applied to the stability analysis of a string of vehicles with a simple form of automatic control.