Constant-sign Lyapunov functionals in the problem of the stability of a functional differential equation

The stability of the zero solution of a non-autonomous functional differential equation of the delayed type is investigated by means of limiting equations and a constant-sign Lyapunov functional, which has a constant-sign derivative. Special cases when the Lyapunov functional and its derivative are explicitly independent of time and the case of an almost periodic equation are also considered. The problem of stabilizing a pendulum in the upper unstable position and the problem of stabilizing the rotational motion of a rigid body are solved as examples.