  In 2002, Fatiha Alabau, Piermarco Cannarsa and Vilmos Komornik investigated
the extent of asymptotic stability of the null solution for weakly coupled
partially damped equations of the second order in time. The main point is that
the damping operator acts only on the first component and, whenever it is
bounded, the coupling is not strong enough to produce an exponential decay in
the energy space associated to the conservative part of the system. As a
consequence, for initial data in the energy space, the rate of decay is not
exponential. Due to the nature of the result it seems at first sight impossible
to obtain the asymptotic stability result by the classical Liapunov method.
Surprisingly enough, this turns out to be possible and we exhibit, under some
compatibility conditions on the operators, an explicit class of Liapunov
functions which allows to do 3 different things: 1) When the problem is reduced
to a stable finite dimensional space, we recover the exponential decay by a
single differential inequality and we estimate the logarithmic decrement of the
solutions with worst (slowest) decay. The estimate is optimal at least for some
values of the parameters.
