  Response lags are generic to almost any physical system and often play a
crucial role in the feedback loops present in artificial nanodevices and
biological molecular machines. In this paper, we perform a comprehensive study
of small stochastic systems governed by an underdamped Langevin equation and
driven out of equilibrium by a time-delayed continuous feedback control. In
their normal operating regime, these systems settle in a nonequilibrium steady
state in which work is permanently extracted from the surrounding heat bath. By
using the Fokker-Planck representation of the dynamics, we derive a set of
second-law-like inequalities that provide bounds to the rate of extracted work.
These inequalities involve additional contributions characterizing the
reduction of entropy production due to the continuous measurement process. We
also show that the non- Markovian nature of the dynamics requires a
modification of the basic relation between dissipation and time-reversal
symmetry breaking. The new relation includes a contribution arising from the
acausal character of the reverse process. This in turn leads to another
second-law-like inequality. We illustrate the general formalism by a detailed
analytical and numerical study of a harmonic oscillator driven by a linear
feedback, which describes actual experimental setups.
