  The article proposes an approach to complete-type and related
Lyapunov-Krasovskii functionals that neither requires knowledge of the
delay-Lyapunov matrix function nor does it involve linear matrix inequalities.
The approach is based on ordinary differential equations (ODEs) that
approximate the time-delay system. The ODEs are derived via spectral methods,
e.g., the Chebyshev collocation method (also called pseudospectral
discretization) or the Legendre tau method. A core insight is that the
Lyapunov-Krasovskii theorem resembles a theorem for Lyapunov-Rumyantsev partial
stability in ODEs. For the linear approximating ODE, only a Lyapunov equation
has to be solved to obtain a partial Lyapunov function. The latter approximates
the Lyapunov-Krasovskii functional. Results are validated by applying
Clenshaw-Curtis and Gauss quadrature to a semi-analytical result of the
functional, yielding a comparable finite-dimensional approximation. In
particular, the article provides a formula for a tight quadratic lower bound,
which is important in applications. Examples confirm that this new bound is
significantly less conservative than known results.
