  Recently, sufficient conditions of stability or instability for time-delay
systems have been proven to be necessary. In this way, a remarkable necessary
and sufficient condition has then been developed by Gomez et al. It is
presented as a simple test of positive definiteness of a matrix issued from the
Lyapunov matrix. In this paper, an extension of this result is presented.
Without going into details, the uniform discretization of the state has been
replaced by projections on the first Legendre polynomials. Like Gomez et al.,
based on convergence arguments, the necessity is obtained in finite order,
which can be calculated analytically. Compared to them, by relying on the fast
convergence rate of Legendre approximation, the required order to ensure
stability has been reduced. Thanks to this major modification, as shown in the
example section, it is possible the find stable regions for low orders and
unstable ones for even smaller orders.
