Stabilization of second-order systems by a relay controller with time delay

We consider a stabilization problem of a system αx" (t) = -x\´(t) + F(x(t), t) + U(t), α = const > 0, h = const > 0, with a discontinuous negative delayed feedback U = -sign x(t - h). Our main conclusion is that, for a bounded derivative F_x(x,t), there are bounded stable solutions, and in the case F(0,t) = 0, there is a feedback U = u(t - h) - sign x(t - h), which provides an exponential decay of x(t)