Stability for trajectories of periodic evolution families in Hilbert spaces
Let $q$ be a positive real number and let $A(\cdot)$ be a $q$-periodic linear operator valued function on a complex Hilbert space $H$, and let $D$ be a dense linear subspace of $H$. Let $\mathcal{U}=\{U(t, s): t\ge s\ge 0\}$ be the evolution family generated by the family $\{A(t)\}$. We prove t...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2014-01-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2014/01/abstr.html |
| _version_ | 1818298939715092480 |
|---|---|
| author | Dorel Barbu Joel Blot Constantin Buse Olivia Saierli |
| author_facet | Dorel Barbu Joel Blot Constantin Buse Olivia Saierli |
| author_sort | Dorel Barbu |
| collection | DOAJ |
| description | Let $q$ be a positive real number and let $A(\cdot)$ be a $q$-periodic
linear operator valued function on a complex Hilbert space $H$, and
let $D$ be a dense linear subspace of $H$.
Let $\mathcal{U}=\{U(t, s): t\ge s\ge 0\}$ be the evolution family
generated by the family $\{A(t)\}$.
We prove that if the solution of the well-posed inhomogeneous
Cauchy Problem
$$\displaylines{
\dot{u}(t) = A(t)u(t)+e^{i\mu t}y, \quad t>0, \cr
u(0) = 0,
}$$
is bounded on ${\mathbb{R}}_+$, for every $y\in D$, and every
$\mu\in\mathbb{R}$, by the positive constant $K\|y\|$,
$K$ being an absolute constant, and if, in addition, for some $x\in D$,
the trajectory $U(\cdot, 0)x$ satisfies a Lipschitz condition on the interval
$(0, q)$, then
$$
\sup_{z\in \mathbb{C}, |z|=1}\sup_{n\in\mathbb{Z}_+}
\|\sum_{k=0}^nz^kU(q, 0)^kx\|:=N(x)<\infty.
$$
The latter discrete boundedness condition has a lot of consequences
concerning the stability of solutions of the abstract nonautonomous
system $\dot u(t)=A(t)u(t)$. To our knowledge, these results are new.
In the special case, when $D=H$ and for every $x\in H $, the map
$U(\cdot, 0)x$ satisfies a Lipschitz condition on the interval $(0, q)$,
the evolution family $\mathcal{U}$ is uniformly exponentially stable.
In the autonomous case, (i.e. when $U(t, s)=U(t-s, 0)$ for every pair
$(t, s)$ with $t\ge s\ge 0$), the latter assumption is too restrictive.
More exactly, in this case, the semigroup $\mathbf{T}:=\{U(t, 0)\}_{t\ge 0}$,
is uniformly continuous. |
| first_indexed | 2024-12-13T04:43:18Z |
| format | Article |
| id | doaj.art-d4071fb677ff46609430bcbd624b9118 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-13T04:43:18Z |
| publishDate | 2014-01-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-d4071fb677ff46609430bcbd624b91182022-12-21T23:59:15ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912014-01-01201401,113Stability for trajectories of periodic evolution families in Hilbert spacesDorel Barbu0Joel Blot1Constantin Buse2Olivia Saierli3 West University of Timisoara, Romania Univ. Paris 1 Sorbonne-Pantheon, Paris Cedex 13, France West University of Timisoara, Romania West University of Timisoara, Romania Let $q$ be a positive real number and let $A(\cdot)$ be a $q$-periodic linear operator valued function on a complex Hilbert space $H$, and let $D$ be a dense linear subspace of $H$. Let $\mathcal{U}=\{U(t, s): t\ge s\ge 0\}$ be the evolution family generated by the family $\{A(t)\}$. We prove that if the solution of the well-posed inhomogeneous Cauchy Problem $$\displaylines{ \dot{u}(t) = A(t)u(t)+e^{i\mu t}y, \quad t>0, \cr u(0) = 0, }$$ is bounded on ${\mathbb{R}}_+$, for every $y\in D$, and every $\mu\in\mathbb{R}$, by the positive constant $K\|y\|$, $K$ being an absolute constant, and if, in addition, for some $x\in D$, the trajectory $U(\cdot, 0)x$ satisfies a Lipschitz condition on the interval $(0, q)$, then $$ \sup_{z\in \mathbb{C}, |z|=1}\sup_{n\in\mathbb{Z}_+} \|\sum_{k=0}^nz^kU(q, 0)^kx\|:=N(x)<\infty. $$ The latter discrete boundedness condition has a lot of consequences concerning the stability of solutions of the abstract nonautonomous system $\dot u(t)=A(t)u(t)$. To our knowledge, these results are new. In the special case, when $D=H$ and for every $x\in H $, the map $U(\cdot, 0)x$ satisfies a Lipschitz condition on the interval $(0, q)$, the evolution family $\mathcal{U}$ is uniformly exponentially stable. In the autonomous case, (i.e. when $U(t, s)=U(t-s, 0)$ for every pair $(t, s)$ with $t\ge s\ge 0$), the latter assumption is too restrictive. More exactly, in this case, the semigroup $\mathbf{T}:=\{U(t, 0)\}_{t\ge 0}$, is uniformly continuous.http://ejde.math.txstate.edu/Volumes/2014/01/abstr.htmlPeriodic evolution families uniform exponential stabilityboundedness strongly continuous semigroupperiodic and almost periodic functions |
| spellingShingle | Dorel Barbu Joel Blot Constantin Buse Olivia Saierli Stability for trajectories of periodic evolution families in Hilbert spaces Electronic Journal of Differential Equations Periodic evolution families uniform exponential stability boundedness strongly continuous semigroup periodic and almost periodic functions |
| title | Stability for trajectories of periodic evolution families in Hilbert spaces |
| title_full | Stability for trajectories of periodic evolution families in Hilbert spaces |
| title_fullStr | Stability for trajectories of periodic evolution families in Hilbert spaces |
| title_full_unstemmed | Stability for trajectories of periodic evolution families in Hilbert spaces |
| title_short | Stability for trajectories of periodic evolution families in Hilbert spaces |
| title_sort | stability for trajectories of periodic evolution families in hilbert spaces |
| topic | Periodic evolution families uniform exponential stability boundedness strongly continuous semigroup periodic and almost periodic functions |
| url | http://ejde.math.txstate.edu/Volumes/2014/01/abstr.html |
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