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We consider a standard discrete-time event-triggered control setting by whicha scheduler collocated with the plant's sensors decides when to transmit sensordata to a remote controller collocated with the plant's actuators. When thescheduler 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 showthat, under mild assumptions, there does not exist a controller and schedulerpair that strictly improves the optimal attenuation bound of periodic controlwith a smaller average transmission rate. Equivalently, given any controllerand scheduler pair, there exists a square-summable disturbance such that eitherthe attenuation bound or the average transmission rate are larger than or equalto those of optimal periodic control.

Event-triggered control cannot improve the $\ell_2$ gain of $h_\infty$ optimal periodic control and transmit at a smaller average rate