Semigroup stability is the long-time decay behavior of a strongly continuous semigroup 
on a Banach space 
. The semigroup has strong
stability if, for every 
,
 |
(1)
|
It has exponential stability if there are constants 
and 
such that, for every 
,
 |
(2)
|
Suppose that the imaginary axis lies in the resolvent set of the infinitesimal
generator 
.
A standard formulation of polynomial stability with decay exponent 
is
 |
(3)
|
(Borichev and Tomilov 2010). Strong stability is pointwise in 
,
whereas exponential stability is uniform
in the operator norm. Uniform operator-norm decay
of 
to zero already implies exponential stability
by the semigroup property. Conditions involving the
operator spectrum and resolvent
set are supplied by the Arendt-Batty-Lyubich-Vũ
theorem, Gearhart-Prüss-Huang
theorem, and Borichev-Tomilov theorem.
