Boundedness and exponential stability for periodic time dependent systems
The time dependent $2$-periodic system \begin{equation*} \dot x{(t)} = A(t)x{(t)} , \ t\in \mathbb{R}, \ \ x(t) \in\mathbb{C}^{n}\tag{A(t)} \end{equation*} is uniformly exponentially stable if and only if for each real number $\mu$ and each $2$-periodic, $\mathbb{C}^{n}$-valued function $f,$ the sol...
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2009-06-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=390 |
| _version_ | 1797830714878590976 |
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| author | Constantin Buse Akbar Zada |
| author_facet | Constantin Buse Akbar Zada |
| author_sort | Constantin Buse |
| collection | DOAJ |
| description | The time dependent $2$-periodic system
\begin{equation*}
\dot x{(t)} = A(t)x{(t)} , \ t\in \mathbb{R}, \ \ x(t) \in\mathbb{C}^{n}\tag{A(t)}
\end{equation*}
is uniformly exponentially stable if and only if for each real number $\mu$ and each $2$-periodic, $\mathbb{C}^{n}$-valued function $f,$ the solution of the Cauchy Problem
\begin{equation*}
\left\{\begin{split}
\dot y{(t)} &= A(t) y{(t)} + e^{i \mu t}f(t),\ \ t\in \mathbb{R}_+, \ y(t) \in \mathbb{C}^{n} \\
y(0) &= 0
\end{split}\right.
\end{equation*}
is bounded. In this note we prove a result that has the above result as an immediate corollary. Some new characterizations for uniform exponential stability of $(A(t))$ in terms of the Datko type theorems are also obtained as corollaries. |
| first_indexed | 2024-04-09T13:41:35Z |
| format | Article |
| id | doaj.art-64e995f1cab54919b3f13b7f6912419a |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:41:35Z |
| publishDate | 2009-06-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-64e995f1cab54919b3f13b7f6912419a2023-05-09T07:52:59ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752009-06-012009371910.14232/ejqtde.2009.1.37390Boundedness and exponential stability for periodic time dependent systemsConstantin Buse0Akbar Zada1West University of Timisoara, Timisoara, RomaniaDepartment of Mathematics, University of Peshawar, Peshawar,PakistanThe time dependent $2$-periodic system \begin{equation*} \dot x{(t)} = A(t)x{(t)} , \ t\in \mathbb{R}, \ \ x(t) \in\mathbb{C}^{n}\tag{A(t)} \end{equation*} is uniformly exponentially stable if and only if for each real number $\mu$ and each $2$-periodic, $\mathbb{C}^{n}$-valued function $f,$ the solution of the Cauchy Problem \begin{equation*} \left\{\begin{split} \dot y{(t)} &= A(t) y{(t)} + e^{i \mu t}f(t),\ \ t\in \mathbb{R}_+, \ y(t) \in \mathbb{C}^{n} \\ y(0) &= 0 \end{split}\right. \end{equation*} is bounded. In this note we prove a result that has the above result as an immediate corollary. Some new characterizations for uniform exponential stability of $(A(t))$ in terms of the Datko type theorems are also obtained as corollaries.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=390 |
| spellingShingle | Constantin Buse Akbar Zada Boundedness and exponential stability for periodic time dependent systems Electronic Journal of Qualitative Theory of Differential Equations |
| title | Boundedness and exponential stability for periodic time dependent systems |
| title_full | Boundedness and exponential stability for periodic time dependent systems |
| title_fullStr | Boundedness and exponential stability for periodic time dependent systems |
| title_full_unstemmed | Boundedness and exponential stability for periodic time dependent systems |
| title_short | Boundedness and exponential stability for periodic time dependent systems |
| title_sort | boundedness and exponential stability for periodic time dependent systems |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=390 |
| work_keys_str_mv | AT constantinbuse boundednessandexponentialstabilityforperiodictimedependentsystems AT akbarzada boundednessandexponentialstabilityforperiodictimedependentsystems |