Fixed points and controllability in delay systems
<p>Schaefer's fixed point theorem is used to study the controllability in an infinite delay system <mml:math alttext="$x'(t)=G(t,x_t)+(Bu)(t)$"> <mml:msup> <mml:mi>x</mml:mi> <mml:mo>&...
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2006-01-01
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| Series: | Fixed Point Theory and Applications |
| Online Access: | http://www.hindawi.com/GetArticle.aspx?doi=10.1155/FPTA/2006/41480 |
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doaj-5d666d05d8fa469b915dcadb5eeace202020-11-25T01:39:16ZengSpringerOpenFixed Point Theory and Applications1687-18202006-01-012006Fixed points and controllability in delay systems<p>Schaefer's fixed point theorem is used to study the controllability in an infinite delay system <mml:math alttext="$x'(t)=G(t,x_t)+(Bu)(t)$"> <mml:msup> <mml:mi>x</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>G</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>B</mml:mi> <mml:mi>u</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math>. A compact map or homotopy is constructed enabling us to show that if there is an <emph>a priori</emph> bound on all possible solutions of the companion control system <mml:math alttext="$x'(t)=lambda[G(t,x_t)+(Bu)(t)]$, $0<lambda<1$"> <mml:msup> <mml:mi>x</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>λ</mml:mi> <mml:mrow> <mml:mo>[</mml:mo> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>B</mml:mi> <mml:mi>u</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:mo>]</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo><</mml:mo> <mml:mi>λ</mml:mi> <mml:mo><</mml:mo> <mml:mn>1</mml:mn> </mml:math>, then there exists a solution for <mml:math alttext="$lambda=1$"> <mml:mi>λ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:math>. The <emph>a priori</emph> bound is established by means of a Liapunov functional or applying an integral inequality. Applications to integral control systems are given to illustrate the approach.</p>http://www.hindawi.com/GetArticle.aspx?doi=10.1155/FPTA/2006/41480 |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| title |
Fixed points and controllability in delay systems |
| spellingShingle |
Fixed points and controllability in delay systems Fixed Point Theory and Applications |
| title_short |
Fixed points and controllability in delay systems |
| title_full |
Fixed points and controllability in delay systems |
| title_fullStr |
Fixed points and controllability in delay systems |
| title_full_unstemmed |
Fixed points and controllability in delay systems |
| title_sort |
fixed points and controllability in delay systems |
| publisher |
SpringerOpen |
| series |
Fixed Point Theory and Applications |
| issn |
1687-1820 |
| publishDate |
2006-01-01 |
| url |
http://www.hindawi.com/GetArticle.aspx?doi=10.1155/FPTA/2006/41480 |
| _version_ |
1715703573721907200 |
| description |
<p>Schaefer's fixed point theorem is used to study the controllability in an infinite delay system <mml:math alttext="$x'(t)=G(t,x_t)+(Bu)(t)$"> <mml:msup> <mml:mi>x</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>G</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>B</mml:mi> <mml:mi>u</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math>. A compact map or homotopy is constructed enabling us to show that if there is an <emph>a priori</emph> bound on all possible solutions of the companion control system <mml:math alttext="$x'(t)=lambda[G(t,x_t)+(Bu)(t)]$, $0<lambda<1$"> <mml:msup> <mml:mi>x</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>λ</mml:mi> <mml:mrow> <mml:mo>[</mml:mo> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>B</mml:mi> <mml:mi>u</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:mo>]</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo><</mml:mo> <mml:mi>λ</mml:mi> <mml:mo><</mml:mo> <mml:mn>1</mml:mn> </mml:math>, then there exists a solution for <mml:math alttext="$lambda=1$"> <mml:mi>λ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:math>. The <emph>a priori</emph> bound is established by means of a Liapunov functional or applying an integral inequality. Applications to integral control systems are given to illustrate the approach.</p> |