Dorel Barbu, Joel Blot, Constantin Buse, Olivia Saierli
Abstract:
Let
be a positive real number and let
be a
-periodic
linear operator valued function on a complex Hilbert space
, and
let
be a dense linear subspace of
.
Let
be the evolution family
generated by the family
.
We prove that if the solution of the well-posed inhomogeneous
Cauchy Problem
is bounded on
,
for every
,
and every
,
by the positive constant
,
being an absolute constant, and if, in addition, for some
,
the trajectory
satisfies a Lipschitz condition on the interval
,
then
The latter discrete boundedness condition has a lot of consequences
concerning the stability of solutions of the abstract nonautonomous
system
.
To our knowledge, these results are new.
In the special case, when
and for every
, the map
satisfies a Lipschitz condition on the interval
,
the evolution family
is uniformly exponentially stable.
In the autonomous case, (i.e. when
for every pair
with
),
the latter assumption is too restrictive.
More exactly, in this case, the semigroup
,
is uniformly continuous.
Submitted August 23, 2013. Published January 3, 2014.
Math Subject Classifications: 47A05, 47A30, 47D06, 47A10, 35B15, 35B10.
Key Words: Periodic evolution families; uniform exponential stability;
boundedness; strongly continuous semigroup;
periodic and almost periodic functions.
Show me the PDF file (285 KB), TEX file, and other files for this article.
Dorel Barbu West University of Timisoara, Department of Mathematics Bd. V. Parvan No. 4, 300223-Timisoara, Romania email: barbu@math.uvt.ro | |
Joel Blot Laboratoire SAMM EA 4543, University Paris 1 Sorbonne-Pantheon, Centre P. M. F. 90 rue de Tolbiac, 75634 Paris Cedex 13, France email: Joel.Blot@univ-paris1.fr | |
Constantin Buse West University of Timisoara, Department of Mathematics Bd. V. Parvan No. 4, 300223-Timisoara, Romania email: buse@math.uvt.ro | |
Olivia Saierli Tibiscus University of Timisoara Department of Computer Science and Applied Informatics Str. Lascar Catargiu, No. 4-6,300559-Timisoara, Romania email: saierli_olivia@yahoo.com |
Return to the EJDE web page