  A subset $A$ of the state space is called uniformly globally weakly
attractive if for any neighborhood $S$ of $A$ and any bounded subset $B$ there
is a uniform finite time $\tau$ so that any trajectory starting in $B$
intersects $S$ within the time not larger than $\tau$. We show that practical
uniform global asymptotic stability (pUGAS) is equivalent to the existence of a
bounded uniformly globally weakly attractive set. This result is valid for a
wide class of distributed parameter systems, including time-delay systems,
switched systems, many classes of PDEs and evolution differential equations in
Banach spaces. We apply our results to show that existence of a non-coercive
Lyapunov function ensures pUGAS for this class of systems. For ordinary
differential equations with uniformly bounded disturbances, the concept of
uniform weak attractivity is equivalent to the well-known notion of weak
attractivity. It is however essentially stronger than weak attractivity for
infinite-dimensional systems, even for linear ones.
