In this paper, we introduce a new policy for the event-based stabilizing controller for linear time-invariant systems. It relies on the definition, in transient time, of a Lyapunov-like function; a positive definite function that vanishes only at the origin. However, whereas a regular Lyapunov function is decaying in time, the Lyapunov-like function is allowed to increase, provided that it remains bounded, in order to get larger inter-event times. An upper threshold is thus fixed in the form of an exponentially decaying function, the rate of decay of which is set by solving a generalized eigenvalue problem. In the vicinity of the steady state, the Lyapunov function is prevented from increasing in time to ensure a faster convergence with less control updates.