  We study a Brownian particle diffusing under a time-modulated stochastic
resetting mechanism to a fixed position. The rate of resetting r(t) is a
function of the time t since the last reset event. We derive a sufficient
condition on r(t) for a steady-state probability distribution of the position
of the particle to exist. We derive the form of the steady-state distributions
under some particular choices of r(t) and also consider the late time
relaxation behavior of the probability distribution. Finally we consider first
passage time properties for the Brownian particle to reach the origin and
derive a formula for the mean first passage time. We study optimal properties
of the mean first passage time and show that a threshold function is at least
locally optimal for the problem of minimizing the mean first passage time.
