Gradient in the Problem of Controlling Processes Described by Linear Pseudohyperbolic Equations
The paper considers the problem of controlling processes whose mathematical model is an initial–boundary value problem for a pseudohyperbolic linear differential equation of high order in the spatial variable and second order in the time variable. The pseudohyperbolic equation is a generalization of...
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Veröffentlicht in: | Differential equations 2024, Vol.60 (2), p.215-226 |
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description | The paper considers the problem of controlling processes whose mathematical model is an initial–boundary value problem for a pseudohyperbolic linear differential equation of high order in the spatial variable and second order in the time variable. The pseudohyperbolic equation is a generalization of the ordinary hyperbolic equation typical in vibration theory. As examples, we consider models of vibrations of moving elastic materials. For the model problems, an energy identity is established and conditions for the uniqueness of a solution are formulated. As an optimization problem, we consider the problem of controlling the right-hand side so as to minimize a quadratic integral functional that evaluates the proximity of the solution to the objective function. From the original functional, a transition is made to a majorant functional, for which the corresponding upper bound is established. An explicit expression for the gradient of this functional is obtained, and adjoint initial–boundary value problems are derived. |
doi_str_mv | 10.1134/S001226612402006X |
format | Article |
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An explicit expression for the gradient of this functional is obtained, and adjoint initial–boundary value problems are derived.</description><identifier>ISSN: 0012-2661</identifier><identifier>EISSN: 1608-3083</identifier><identifier>DOI: 10.1134/S001226612402006X</identifier><language>eng</language><publisher>Moscow: Pleiades Publishing</publisher><subject>Boundary value problems ; Control Theory ; Difference and Functional Equations ; Differential equations ; Mathematical models ; Mathematics ; Mathematics and Statistics ; Optimization ; Ordinary Differential Equations ; Partial Differential Equations ; Upper bounds</subject><ispartof>Differential equations, 2024, Vol.60 (2), p.215-226</ispartof><rights>Pleiades Publishing, Ltd. 2024</rights><rights>Pleiades Publishing, Ltd. 2024.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-p156t-776cf077b77573afc3daea943585a664d69dcbb10f6da02b54b4fe94e13a476a3</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/S001226612402006X$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1134/S001226612402006X$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>315,781,785,27926,27927,41490,42559,51321</link.rule.ids></links><search><creatorcontrib>Romanenkov, A. M.</creatorcontrib><title>Gradient in the Problem of Controlling Processes Described by Linear Pseudohyperbolic Equations</title><title>Differential equations</title><addtitle>Diff Equat</addtitle><description>The paper considers the problem of controlling processes whose mathematical model is an initial–boundary value problem for a pseudohyperbolic linear differential equation of high order in the spatial variable and second order in the time variable. The pseudohyperbolic equation is a generalization of the ordinary hyperbolic equation typical in vibration theory. As examples, we consider models of vibrations of moving elastic materials. For the model problems, an energy identity is established and conditions for the uniqueness of a solution are formulated. As an optimization problem, we consider the problem of controlling the right-hand side so as to minimize a quadratic integral functional that evaluates the proximity of the solution to the objective function. From the original functional, a transition is made to a majorant functional, for which the corresponding upper bound is established. An explicit expression for the gradient of this functional is obtained, and adjoint initial–boundary value problems are derived.</description><subject>Boundary value problems</subject><subject>Control Theory</subject><subject>Difference and Functional Equations</subject><subject>Differential equations</subject><subject>Mathematical models</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Optimization</subject><subject>Ordinary Differential Equations</subject><subject>Partial Differential Equations</subject><subject>Upper bounds</subject><issn>0012-2661</issn><issn>1608-3083</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><recordid>eNplkEtLw0AUhQdRsD5-gLsB19E772QptVahYEEFd2EmubEpcSadSRb99zZUcOHqwjkf53APITcM7hgT8v4NgHGuNeMSOID-PCEzpiHPBOTilMwmO5v8c3KR0hYACsPUjJTLaOsW_UBbT4cN0nUMrsNvGho6D36Ioeta_zXJFaaEiT5iqmLrsKZuT1etRxvpOuFYh82-x-hC11Z0sRvt0AafrshZY7uE17_3knw8Ld7nz9nqdfkyf1hlPVN6yIzRVQPGOGOUEbapRG3RFlKoXFmtZa2LunKOQaNrC9wp6WSDhUQmrDTaiktye8ztY9iNmIZyG8boD5WlAC2LXGnJDxQ_UqmPh68w_lEMymnI8t-Q4gecAGaq</recordid><startdate>2024</startdate><enddate>2024</enddate><creator>Romanenkov, A. M.</creator><general>Pleiades Publishing</general><general>Springer Nature B.V</general><scope>7SC</scope><scope>7TB</scope><scope>8FD</scope><scope>FR3</scope><scope>JQ2</scope><scope>KR7</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>2024</creationdate><title>Gradient in the Problem of Controlling Processes Described by Linear Pseudohyperbolic Equations</title><author>Romanenkov, A. M.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-p156t-776cf077b77573afc3daea943585a664d69dcbb10f6da02b54b4fe94e13a476a3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Boundary value problems</topic><topic>Control Theory</topic><topic>Difference and Functional Equations</topic><topic>Differential equations</topic><topic>Mathematical models</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Optimization</topic><topic>Ordinary Differential Equations</topic><topic>Partial Differential Equations</topic><topic>Upper bounds</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Romanenkov, A. M.</creatorcontrib><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Civil Engineering Abstracts</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Differential equations</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Romanenkov, A. M.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Gradient in the Problem of Controlling Processes Described by Linear Pseudohyperbolic Equations</atitle><jtitle>Differential equations</jtitle><stitle>Diff Equat</stitle><date>2024</date><risdate>2024</risdate><volume>60</volume><issue>2</issue><spage>215</spage><epage>226</epage><pages>215-226</pages><issn>0012-2661</issn><eissn>1608-3083</eissn><abstract>The paper considers the problem of controlling processes whose mathematical model is an initial–boundary value problem for a pseudohyperbolic linear differential equation of high order in the spatial variable and second order in the time variable. The pseudohyperbolic equation is a generalization of the ordinary hyperbolic equation typical in vibration theory. As examples, we consider models of vibrations of moving elastic materials. For the model problems, an energy identity is established and conditions for the uniqueness of a solution are formulated. As an optimization problem, we consider the problem of controlling the right-hand side so as to minimize a quadratic integral functional that evaluates the proximity of the solution to the objective function. From the original functional, a transition is made to a majorant functional, for which the corresponding upper bound is established. An explicit expression for the gradient of this functional is obtained, and adjoint initial–boundary value problems are derived.</abstract><cop>Moscow</cop><pub>Pleiades Publishing</pub><doi>10.1134/S001226612402006X</doi><tpages>12</tpages></addata></record> |
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subjects | Boundary value problems Control Theory Difference and Functional Equations Differential equations Mathematical models Mathematics Mathematics and Statistics Optimization Ordinary Differential Equations Partial Differential Equations Upper bounds |
title | Gradient in the Problem of Controlling Processes Described by Linear Pseudohyperbolic Equations |
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