  We consider a standard discrete-time event-triggered control setting by which
a scheduler collocated with the plant's sensors decides when to transmit sensor
data to a remote controller collocated with the plant's actuators. When the
scheduler transmits periodically with period larger than or equal to one, the
$h_\infty$ optimal controller guarantees an optimal attenuation bound ($\ell_2$
gain) from any square-summable disturbance input to a plant's output. We show
that, under mild assumptions, there does not exist a controller and scheduler
pair that strictly improves the optimal attenuation bound of periodic control
with a smaller average transmission rate. Equivalently, given any controller
and scheduler pair, there exists a square-summable disturbance such that either
the attenuation bound or the average transmission rate are larger than or equal
to those of optimal periodic control.
