On boundary control problems for the Klein–Gordon–Fock equation with an integrable coefficient
We consider the process described in the rectangle Q T = [0 ≤ x ≤ l ] × [0 ≤ t ≤ T ] by the equation u tt - u xx - q ( x, t ) u = 0 with the condition u ( l, t ) = 0, where the coefficient q ( x, t ) is only square integrable on Q T . We show that for T = 2 l the problem of boundary control of this...
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| Veröffentlicht in: | Differential equations 2015-05, Vol.51 (5), p.701-709 |
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| container_title | Differential equations |
| container_volume | 51 |
| creator | Kritskov, L. V. |
| description | We consider the process described in the rectangle
Q
T
= [0 ≤
x
≤
l
] × [0 ≤
t
≤
T
] by the equation
u
tt
-
u
xx
-
q
(
x, t
)
u
= 0 with the condition
u
(
l, t
) = 0, where the coefficient
q
(
x, t
) is only square integrable on
Q
T
. We show that for
T
= 2
l
the problem of boundary control of this process by the condition
u
(0,
t
) = µ(
t
) has exactly one solution in the class
W
2
1
(
Q
T
) under minimum requirements on the smoothness of the initial and terminal functions and under natural matching conditions at
x
=
l
. |
| doi_str_mv | 10.1134/S0012266115050122 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_miscellaneous_1718918782</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>3719687071</sourcerecordid><originalsourceid>FETCH-LOGICAL-c349t-a864b519156eab32d776fd8a62c7ceff1ea0a20a522399215176d0e0449838ae3</originalsourceid><addsrcrecordid>eNp1kMFOwzAQRC0EEqXwAdwsceES8NqJ4xxRRQuiUg_AOXKSTZuS2q2dCHHjH_hDvgRH5YBAnHakfTMaDSHnwK4ARHz9yBhwLiVAwpJBHpARSKYiwZQ4JKPhHQ3_Y3Li_ZoxlqWQjEixMLSwvam0e6OlNZ2zLd06W7S48bS2jnYrpA8tNubz_WNmXWUHMbXlC8Vdr7vGGvradCuqDW1Mh0ungzdEYV03ZYOmOyVHtW49nn3fMXme3j5N7qL5YnY_uZlHpYizLtJKxkUCGSQSdSF4laayrpSWvEzLEAaomeZMJ5yLLOOQQCorhiyOMyWURjEml_vcUH_Xo-_yTeNLbFtt0PY-hxRUBipVPKAXv9C17Z0J7XKQKlOMAxeBgj1VOuu9wzrfumYThsqB5cPq-Z_Vg4fvPT6wZonuR_K_pi9Yk4T4</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>1689802123</pqid></control><display><type>article</type><title>On boundary control problems for the Klein–Gordon–Fock equation with an integrable coefficient</title><source>SpringerLink Journals - AutoHoldings</source><creator>Kritskov, L. V.</creator><creatorcontrib>Kritskov, L. V.</creatorcontrib><description>We consider the process described in the rectangle
Q
T
= [0 ≤
x
≤
l
] × [0 ≤
t
≤
T
] by the equation
u
tt
-
u
xx
-
q
(
x, t
)
u
= 0 with the condition
u
(
l, t
) = 0, where the coefficient
q
(
x, t
) is only square integrable on
Q
T
. We show that for
T
= 2
l
the problem of boundary control of this process by the condition
u
(0,
t
) = µ(
t
) has exactly one solution in the class
W
2
1
(
Q
T
) under minimum requirements on the smoothness of the initial and terminal functions and under natural matching conditions at
x
=
l
.</description><identifier>ISSN: 0012-2661</identifier><identifier>EISSN: 1608-3083</identifier><identifier>DOI: 10.1134/S0012266115050122</identifier><language>eng</language><publisher>Moscow: Pleiades Publishing</publisher><subject>Boundary conditions ; Boundary control ; Coefficients ; Control Theory ; Difference and Functional Equations ; Differential equations ; Integrals ; Matching ; Mathematical analysis ; Mathematics ; Mathematics and Statistics ; Ordinary Differential Equations ; Partial Differential Equations ; Rectangles ; Smoothness ; Studies ; Terminals</subject><ispartof>Differential equations, 2015-05, Vol.51 (5), p.701-709</ispartof><rights>Pleiades Publishing, Ltd. 2015</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c349t-a864b519156eab32d776fd8a62c7ceff1ea0a20a522399215176d0e0449838ae3</citedby><cites>FETCH-LOGICAL-c349t-a864b519156eab32d776fd8a62c7ceff1ea0a20a522399215176d0e0449838ae3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1134/S0012266115050122$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1134/S0012266115050122$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27903,27904,41467,42536,51297</link.rule.ids></links><search><creatorcontrib>Kritskov, L. V.</creatorcontrib><title>On boundary control problems for the Klein–Gordon–Fock equation with an integrable coefficient</title><title>Differential equations</title><addtitle>Diff Equat</addtitle><description>We consider the process described in the rectangle
Q
T
= [0 ≤
x
≤
l
] × [0 ≤
t
≤
T
] by the equation
u
tt
-
u
xx
-
q
(
x, t
)
u
= 0 with the condition
u
(
l, t
) = 0, where the coefficient
q
(
x, t
) is only square integrable on
Q
T
. We show that for
T
= 2
l
the problem of boundary control of this process by the condition
u
(0,
t
) = µ(
t
) has exactly one solution in the class
W
2
1
(
Q
T
) under minimum requirements on the smoothness of the initial and terminal functions and under natural matching conditions at
x
=
l
.</description><subject>Boundary conditions</subject><subject>Boundary control</subject><subject>Coefficients</subject><subject>Control Theory</subject><subject>Difference and Functional Equations</subject><subject>Differential equations</subject><subject>Integrals</subject><subject>Matching</subject><subject>Mathematical analysis</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Ordinary Differential Equations</subject><subject>Partial Differential Equations</subject><subject>Rectangles</subject><subject>Smoothness</subject><subject>Studies</subject><subject>Terminals</subject><issn>0012-2661</issn><issn>1608-3083</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2015</creationdate><recordtype>article</recordtype><sourceid>8G5</sourceid><sourceid>BENPR</sourceid><sourceid>GUQSH</sourceid><sourceid>M2O</sourceid><recordid>eNp1kMFOwzAQRC0EEqXwAdwsceES8NqJ4xxRRQuiUg_AOXKSTZuS2q2dCHHjH_hDvgRH5YBAnHakfTMaDSHnwK4ARHz9yBhwLiVAwpJBHpARSKYiwZQ4JKPhHQ3_Y3Li_ZoxlqWQjEixMLSwvam0e6OlNZ2zLd06W7S48bS2jnYrpA8tNubz_WNmXWUHMbXlC8Vdr7vGGvradCuqDW1Mh0ungzdEYV03ZYOmOyVHtW49nn3fMXme3j5N7qL5YnY_uZlHpYizLtJKxkUCGSQSdSF4laayrpSWvEzLEAaomeZMJ5yLLOOQQCorhiyOMyWURjEml_vcUH_Xo-_yTeNLbFtt0PY-hxRUBipVPKAXv9C17Z0J7XKQKlOMAxeBgj1VOuu9wzrfumYThsqB5cPq-Z_Vg4fvPT6wZonuR_K_pi9Yk4T4</recordid><startdate>20150501</startdate><enddate>20150501</enddate><creator>Kritskov, L. V.</creator><general>Pleiades Publishing</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>04Q</scope><scope>04W</scope><scope>3V.</scope><scope>7SC</scope><scope>7TB</scope><scope>7WY</scope><scope>7WZ</scope><scope>7XB</scope><scope>87Z</scope><scope>8AL</scope><scope>8FD</scope><scope>8FE</scope><scope>8FG</scope><scope>8FK</scope><scope>8FL</scope><scope>8G5</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>ARAPS</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BEZIV</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>FR3</scope><scope>FRNLG</scope><scope>F~G</scope><scope>GNUQQ</scope><scope>GUQSH</scope><scope>HCIFZ</scope><scope>JQ2</scope><scope>K60</scope><scope>K6~</scope><scope>K7-</scope><scope>KR7</scope><scope>L.-</scope><scope>L6V</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><scope>M0C</scope><scope>M0N</scope><scope>M2O</scope><scope>M7S</scope><scope>MBDVC</scope><scope>P5Z</scope><scope>P62</scope><scope>PADUT</scope><scope>PQBIZ</scope><scope>PQBZA</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PTHSS</scope><scope>PYYUZ</scope><scope>Q9U</scope></search><sort><creationdate>20150501</creationdate><title>On boundary control problems for the Klein–Gordon–Fock equation with an integrable coefficient</title><author>Kritskov, L. V.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c349t-a864b519156eab32d776fd8a62c7ceff1ea0a20a522399215176d0e0449838ae3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2015</creationdate><topic>Boundary conditions</topic><topic>Boundary control</topic><topic>Coefficients</topic><topic>Control Theory</topic><topic>Difference and Functional Equations</topic><topic>Differential equations</topic><topic>Integrals</topic><topic>Matching</topic><topic>Mathematical analysis</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Ordinary Differential Equations</topic><topic>Partial Differential Equations</topic><topic>Rectangles</topic><topic>Smoothness</topic><topic>Studies</topic><topic>Terminals</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Kritskov, L. V.</creatorcontrib><collection>CrossRef</collection><collection>India Database</collection><collection>India Database: Science & Technology</collection><collection>ProQuest Central (Corporate)</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>ABI/INFORM Collection</collection><collection>ABI/INFORM Global (PDF only)</collection><collection>ProQuest Central (purchase pre-March 2016)</collection><collection>ABI/INFORM Global (Alumni Edition)</collection><collection>Computing Database (Alumni Edition)</collection><collection>Technology Research Database</collection><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>ProQuest Central (Alumni) (purchase pre-March 2016)</collection><collection>ABI/INFORM Collection (Alumni Edition)</collection><collection>Research Library (Alumni Edition)</collection><collection>Materials Science & Engineering Collection</collection><collection>ProQuest Central (Alumni Edition)</collection><collection>ProQuest Central UK/Ireland</collection><collection>Advanced Technologies & Aerospace Collection</collection><collection>ProQuest Central Essentials</collection><collection>ProQuest Central</collection><collection>Business Premium Collection</collection><collection>Technology Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central Korea</collection><collection>Engineering Research Database</collection><collection>Business Premium Collection (Alumni)</collection><collection>ABI/INFORM Global (Corporate)</collection><collection>ProQuest Central Student</collection><collection>Research Library Prep</collection><collection>SciTech Premium Collection</collection><collection>ProQuest Computer Science Collection</collection><collection>ProQuest Business Collection (Alumni Edition)</collection><collection>ProQuest Business Collection</collection><collection>Computer Science Database</collection><collection>Civil Engineering Abstracts</collection><collection>ABI/INFORM Professional Advanced</collection><collection>ProQuest Engineering Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><collection>ABI/INFORM Global</collection><collection>Computing Database</collection><collection>Research Library</collection><collection>Engineering Database</collection><collection>Research Library (Corporate)</collection><collection>Advanced Technologies & Aerospace Database</collection><collection>ProQuest Advanced Technologies & Aerospace Collection</collection><collection>Research Library China</collection><collection>ProQuest One Business</collection><collection>ProQuest One Business (Alumni)</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>Engineering Collection</collection><collection>ABI/INFORM Collection China</collection><collection>ProQuest Central Basic</collection><jtitle>Differential equations</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Kritskov, L. V.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On boundary control problems for the Klein–Gordon–Fock equation with an integrable coefficient</atitle><jtitle>Differential equations</jtitle><stitle>Diff Equat</stitle><date>2015-05-01</date><risdate>2015</risdate><volume>51</volume><issue>5</issue><spage>701</spage><epage>709</epage><pages>701-709</pages><issn>0012-2661</issn><eissn>1608-3083</eissn><abstract>We consider the process described in the rectangle
Q
T
= [0 ≤
x
≤
l
] × [0 ≤
t
≤
T
] by the equation
u
tt
-
u
xx
-
q
(
x, t
)
u
= 0 with the condition
u
(
l, t
) = 0, where the coefficient
q
(
x, t
) is only square integrable on
Q
T
. We show that for
T
= 2
l
the problem of boundary control of this process by the condition
u
(0,
t
) = µ(
t
) has exactly one solution in the class
W
2
1
(
Q
T
) under minimum requirements on the smoothness of the initial and terminal functions and under natural matching conditions at
x
=
l
.</abstract><cop>Moscow</cop><pub>Pleiades Publishing</pub><doi>10.1134/S0012266115050122</doi><tpages>9</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0012-2661 |
| ispartof | Differential equations, 2015-05, Vol.51 (5), p.701-709 |
| issn | 0012-2661 1608-3083 |
| language | eng |
| recordid | cdi_proquest_miscellaneous_1718918782 |
| source | SpringerLink Journals - AutoHoldings |
| subjects | Boundary conditions Boundary control Coefficients Control Theory Difference and Functional Equations Differential equations Integrals Matching Mathematical analysis Mathematics Mathematics and Statistics Ordinary Differential Equations Partial Differential Equations Rectangles Smoothness Studies Terminals |
| title | On boundary control problems for the Klein–Gordon–Fock equation with an integrable coefficient |
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