Optimal Control Laws for Batch and Semi-batch Reactors Using the Concept of Extents

<html xmlns:v="urn:schemas-microsoft-com:vml" xmlns:o="urn:schemas-microsoft-com:office:office" xmlns:w="urn:schemas-microsoft-com:office:word" xmlns:m="http://schemas.microsoft.com/office/2004/12/omml" xmlns:mv="http://macVmlSchemaUri" xmlns="http://www.w3.org/TR/REC-html40"> <head> <meta http-equiv=Content-Type content="text/html; charset=utf-8"> <meta name=ProgId content=Word.Document> <meta name=Generator content="Microsoft Word 14"> <meta name=Originator content="Microsoft Word 14"> <link rel=File-List href="AIChE_abstract_files/filelist.xml"> <!--[if gte mso 9]><xml> <o:DocumentProperties> <o:LastAuthor>Diogo Filipe Mateus Rodrigues</o:LastAuthor> <o:Revision>2</o:Revision> <o:TotalTime>1</o:TotalTime> <o:LastPrinted>2017-04-16T13:39:00Z</o:LastPrinted> <o:Created>2017-04-16T13:40:00Z</o:Created> <o:LastSaved>2017-04-16T13:40:00Z</o:LastSaved> <o:Pages>1</o:Pages> <o:Words>1261</o:Words> <o:Characters>7191</o:Characters> <o:Company>EPFL</o:Company> <o:Lines>59</o:Lines> <o:Paragraphs>16</o:Paragraphs> <o:CharactersWithSpaces>8436</o:CharactersWithSpaces> <o:Version>14.0</o:Version> </o:DocumentProperties> </xml><![endif]--> <link rel=themeData href="AIChE_abstract_files/themedata.xml"> <!--[if gte mso 9]><xml> <w:WordDocument> <w:Zoom>196</w:Zoom> <w:TrackMoves/> <w:TrackFormatting/> <w:PunctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:DoNotPromoteQF/> <w:LidThemeOther>EN-US</w:LidThemeOther> <w:LidThemeAsian>JA</w:LidThemeAsian> <w:LidThemeComplexScript>X-NONE</w:LidThemeComplexScript> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w:DontGrowAutofit/> <w:SplitPgBreakAndParaMark/> <w:UseFELayout/> </w:Compatibility> <m:mathPr> <m:mathFont m:val="Cambria Math"/> <m:brkBin m:val="before"/> <m:brkBinSub m:val="&#45;-"/> <m:smallFrac m:val="off"/> <m:dispDef/> <m:lMargin m:val="0"/> <m:rMargin m:val="0"/> <m:defJc m:val="centerGroup"/> <m:wrapIndent m:val="1440"/> <m:intLim m:val="subSup"/> <m:naryLim m:val="undOvr"/> </m:mathPr></w:WordDocument> </xml><![endif]--><!--[if gte mso 9]><xml> <w:LatentStyles DefLockedState="false" DefUnhideWhenUsed="true" DefSemiHidden="true" DefQFormat="false" DefPriority="99" LatentStyleCount="276"> <w:LsdException Locked="false" Priority="0" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Normal"/> <w:LsdException Locked="false" Priority="9" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="heading 1"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 2"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 3"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 4"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 5"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 6"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 7"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 8"/> <w:LsdException Locked="false" Priority="9" QFormat="true" Name="heading 9"/> <w:LsdException Locked="false" Priority="39" Name="toc 1"/> <w:LsdException Locked="false" Priority="39" Name="toc 2"/> <w:LsdException Locked="false" Priority="39" Name="toc 3"/> <w:LsdException Locked="false" Priority="39" Name="toc 4"/> <w:LsdException Locked="false" Priority="39" Name="toc 5"/> <w:LsdException Locked="false" Priority="39" Name="toc 6"/> <w:LsdException Locked="false" Priority="39" Name="toc 7"/> <w:LsdException Locked="false" Priority="39" Name="toc 8"/> <w:LsdException Locked="false" Priority="39" Name="toc 9"/> <w:LsdException Locked="false" Priority="35" QFormat="true" Name="caption"/> <w:LsdException Locked="false" Priority="10" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Title"/> <w:LsdException Locked="false" Priority="1" Name="Default Paragraph Font"/> <w:LsdException Locked="false" Priority="11" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Subtitle"/> <w:LsdException Locked="false" Priority="22" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Strong"/> <w:LsdException Locked="false" Priority="20" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Emphasis"/> <w:LsdException Locked="false" Priority="59" SemiHidden="false" UnhideWhenUsed="false" Name="Table Grid"/> <w:LsdException Locked="false" UnhideWhenUsed="false" Name="Placeholder Text"/> <w:LsdException Locked="false" Priority="1" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="No Spacing"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false" UnhideWhenUsed="false" Name="Light Shading"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false" UnhideWhenUsed="false" Name="Light List"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false" UnhideWhenUsed="false" Name="Light Grid"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 1"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 2"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 1"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 2"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 1"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 2"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 3"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false" UnhideWhenUsed="false" Name="Dark List"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Shading"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful List"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Grid"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false" UnhideWhenUsed="false" Name="Light Shading Accent 1"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false" UnhideWhenUsed="false" Name="Light List Accent 1"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false" UnhideWhenUsed="false" Name="Light Grid Accent 1"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 1 Accent 1"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 2 Accent 1"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 1 Accent 1"/> <w:LsdException Locked="false" UnhideWhenUsed="false" Name="Revision"/> <w:LsdException Locked="false" Priority="34" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="List Paragraph"/> <w:LsdException Locked="false" Priority="29" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Quote"/> <w:LsdException Locked="false" Priority="30" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Intense Quote"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 2 Accent 1"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 1 Accent 1"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 2 Accent 1"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 3 Accent 1"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false" UnhideWhenUsed="false" Name="Dark List Accent 1"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Shading Accent 1"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful List Accent 1"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Grid Accent 1"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false" UnhideWhenUsed="false" Name="Light Shading Accent 2"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false" UnhideWhenUsed="false" Name="Light List Accent 2"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false" UnhideWhenUsed="false" Name="Light Grid Accent 2"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 1 Accent 2"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 2 Accent 2"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 1 Accent 2"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 2 Accent 2"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 1 Accent 2"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 2 Accent 2"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 3 Accent 2"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false" UnhideWhenUsed="false" Name="Dark List Accent 2"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Shading Accent 2"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful List Accent 2"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Grid Accent 2"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false" UnhideWhenUsed="false" Name="Light Shading Accent 3"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false" UnhideWhenUsed="false" Name="Light List Accent 3"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false" UnhideWhenUsed="false" Name="Light Grid Accent 3"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 1 Accent 3"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 2 Accent 3"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 1 Accent 3"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 2 Accent 3"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 1 Accent 3"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 2 Accent 3"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 3 Accent 3"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false" UnhideWhenUsed="false" Name="Dark List Accent 3"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Shading Accent 3"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful List Accent 3"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Grid Accent 3"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false" UnhideWhenUsed="false" Name="Light Shading Accent 4"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false" UnhideWhenUsed="false" Name="Light List Accent 4"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false" UnhideWhenUsed="false" Name="Light Grid Accent 4"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 1 Accent 4"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 2 Accent 4"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 1 Accent 4"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 2 Accent 4"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 1 Accent 4"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 2 Accent 4"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 3 Accent 4"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false" UnhideWhenUsed="false" Name="Dark List Accent 4"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Shading Accent 4"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful List Accent 4"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Grid Accent 4"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false" UnhideWhenUsed="false" Name="Light Shading Accent 5"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false" UnhideWhenUsed="false" Name="Light List Accent 5"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false" UnhideWhenUsed="false" Name="Light Grid Accent 5"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 1 Accent 5"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 2 Accent 5"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 1 Accent 5"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 2 Accent 5"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 1 Accent 5"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 2 Accent 5"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 3 Accent 5"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false" UnhideWhenUsed="false" Name="Dark List Accent 5"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Shading Accent 5"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful List Accent 5"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Grid Accent 5"/> <w:LsdException Locked="false" Priority="60" SemiHidden="false" UnhideWhenUsed="false" Name="Light Shading Accent 6"/> <w:LsdException Locked="false" Priority="61" SemiHidden="false" UnhideWhenUsed="false" Name="Light List Accent 6"/> <w:LsdException Locked="false" Priority="62" SemiHidden="false" UnhideWhenUsed="false" Name="Light Grid Accent 6"/> <w:LsdException Locked="false" Priority="63" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 1 Accent 6"/> <w:LsdException Locked="false" Priority="64" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Shading 2 Accent 6"/> <w:LsdException Locked="false" Priority="65" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 1 Accent 6"/> <w:LsdException Locked="false" Priority="66" SemiHidden="false" UnhideWhenUsed="false" Name="Medium List 2 Accent 6"/> <w:LsdException Locked="false" Priority="67" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 1 Accent 6"/> <w:LsdException Locked="false" Priority="68" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 2 Accent 6"/> <w:LsdException Locked="false" Priority="69" SemiHidden="false" UnhideWhenUsed="false" Name="Medium Grid 3 Accent 6"/> <w:LsdException Locked="false" Priority="70" SemiHidden="false" UnhideWhenUsed="false" Name="Dark List Accent 6"/> <w:LsdException Locked="false" Priority="71" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Shading Accent 6"/> <w:LsdException Locked="false" Priority="72" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful List Accent 6"/> <w:LsdException Locked="false" Priority="73" SemiHidden="false" UnhideWhenUsed="false" Name="Colorful Grid Accent 6"/> <w:LsdException Locked="false" Priority="19" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Subtle Emphasis"/> <w:LsdException Locked="false" Priority="21" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Intense Emphasis"/> <w:LsdException Locked="false" Priority="31" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Subtle Reference"/> <w:LsdException Locked="false" Priority="32" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Intense Reference"/> <w:LsdException Locked="false" Priority="33" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Book Title"/> <w:LsdException Locked="false" Priority="37" Name="Bibliography"/> <w:LsdException Locked="false" Priority="39" QFormat="true" Name="TOC Heading"/> </w:LatentStyles> </xml><![endif]--> <style> <!-- /* Font Definitions */ @font-face {font-family:"Arial Unicode MS"; panose-1:2 11 6 4 2 2 2 2 2 4; mso-font-charset:0; mso-generic-font-family:auto; mso-font-pitch:variable; mso-font-signature:-134238209 -371195905 63 0 4129279 0;} /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-unhide:no; mso-style-parent:""; margin:0cm; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman"; mso-fareast-font-family:"Arial Unicode MS"; border:none;} a:link, span.MsoHyperlink {mso-style-unhide:no; mso-style-parent:""; text-decoration:underline; text-underline:single;} a:visited, span.MsoHyperlinkFollowed {mso-style-noshow:yes; mso-style-priority:99; color:fuchsia; mso-themecolor:followedhyperlink; text-decoration:underline; text-underline:single;} p.HeaderFooter, li.HeaderFooter, div.HeaderFooter {mso-style-name:"Header & Footer"; mso-style-unhide:no; mso-style-parent:""; margin:0cm; margin-bottom:.0001pt; mso-pagination:widow-orphan; tab-stops:right 451.0pt; font-size:12.0pt; font-family:Helvetica; mso-fareast-font-family:"Arial Unicode MS"; mso-bidi-font-family:"Arial Unicode MS"; color:black; border:none;} p.Default, li.Default, div.Default {mso-style-name:Default; mso-style-unhide:no; mso-style-parent:""; margin:0cm; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:11.0pt; font-family:Helvetica; mso-fareast-font-family:"Arial Unicode MS"; mso-bidi-font-family:"Arial Unicode MS"; color:black; border:none;} span.Hyperlink0 {mso-style-name:"Hyperlink\.0"; mso-style-unhide:no; mso-style-parent:Hyperlink; text-decoration:underline; text-underline:single;} .MsoChpDefault {mso-style-type:export-only; mso-default-props:yes; font-size:10.0pt; mso-ansi-font-size:10.0pt; mso-bidi-font-size:10.0pt; mso-fareast-font-family:"Arial Unicode MS"; border:none;} .MsoPapDefault {mso-style-type:export-only;} /* Page Definitions */ @page {mso-footnote-separator:url(":AIChE_abstract_files:header.htm") fs; mso-footnote-continuation-separator:url(":AIChE_abstract_files:header.htm") fcs; mso-endnote-separator:url(":AIChE_abstract_files:header.htm") es; mso-endnote-continuation-separator:url(":AIChE_abstract_files:header.htm") ecs;} @page WordSection1 {size:595.0pt 842.0pt; margin:72.0pt 90.0pt 72.0pt 90.0pt; mso-header-margin:35.4pt; mso-footer-margin:35.4pt; mso-header:url(":AIChE_abstract_files:header.htm") h1; mso-footer:url(":AIChE_abstract_files:header.htm") f1; mso-paper-source:0;} div.WordSection1 {page:WordSection1;} --> </style> <!--[if gte mso 10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; border:none;} </style> <![endif]--><!--[if gte mso 9]><xml> <o:shapedefaults v:ext="edit" spidmax="1026"/> </xml><![endif]--><!--[if gte mso 9]><xml> <o:shapelayout v:ext="edit"> <o:idmap v:ext="edit" data="1"/> </o:shapelayout></xml><![endif]--> </head> <body lang=EN-US link="#000000" vlink=fuchsia style='tab-interval:36.0pt'> <div class=WordSection1> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>In the context of dynamic optimization of batch and semi-batch reactors, this contribution presents a method that uses the concept of extents to generate solutions that satisfy the necessary conditions </span><span lang=FR style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:FR'>of </span><span style='font-size:12.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>optimality given by Pontryagin</span><span lang=FR style='font-size:12.0pt;font-family: "Times New Roman";mso-bidi-font-family:"Arial Unicode MS";mso-ansi-language: FR'>’s</span><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-bidi-font-family:"Arial Unicode MS"'> maximum principle. The method is divided in two parts. In the first part, the reactor model is written in terms of decoupled extents, and adjoint-free optimal control laws are generated for all possible types of arcs that may occur in the optimal solution. In the second part, the correct sequence of arcs is determined, and, for each sequence, the optimal switching times and initial conditions are computed numerically.</span><span style='font-size:12.0pt; font-family:"Times New Roman";mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 106.35pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>The optimal control problems (OCPs) are formulated in Mayer form, with <i>n<sub>u</sub></i> piecewise-continuous inputs <b>u</b>(<i>t</i>), <i>n<sub>x</sub></i> states <b>x</b>(<i>t</i>) described by the differential equations <b>ẋ</b>(<i>t</i>) = <b>f</b>(<b>x</b>(<i>t</i>),<b>u</b>(<i>t</i>)), <b>x</b>(<i>t</i><sub>0</sub>) = <b>x</b><sub>0</sub>, the cost function </span><i><span lang=FR style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:FR'>φ</span></i><span style='font-size: 12.0pt;font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>(<i>t<sub>f</sub></i>,<b>x</b>(<i>t<sub>f</sub></i>)), the <i>n<sub>t</sub></i> terminal constraints </span><b><span lang=FR style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:FR'>ψ</span></b><span style='font-size: 12.0pt;font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>(<i>t<sub>f</sub></i>,<b>x</b>(<i>t<sub>f</sub></i>)) ≤ <b>0</b>, the <i>n<sub>g</sub></i> mixed path constraints <b>g</b>(<b>x</b>(<i>t</i>),<b>u</b>(<i>t</i>)) ≤ <b>0</b> and the <i>n<sub>h</sub></i> first-order pure-state constraints <b>h</b>(<b>x</b>(<i>t</i>)) ≤ <b>0</b>. Note that this class of OCPs is not restrictive in most dynamic optimization problems dealing with reactors.</span><span style='font-size:12.0pt;font-family: "Times New Roman";mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>The optimal input trajectories are composed of a (typically finite) number of arcs. For each arc and for each input, the optimal input is determined by either an active path constraint or a condition that expresses a physical compromise that depends exclusively on the dynamics of the system [1].</span><span style='font-size:12.0pt;font-family:"Times New Roman";mso-fareast-font-family: "Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>Let the input <i>u<sub>j</sub></i> be one element of <b>u</b>. The goal is to find an expression that relates the optimal input <i>u<sub>j</sub></i>, or one of its time derivatives, to the states, the inputs or the time derivatives of the inputs, thus resulting in an </span><i><span lang=FR style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:FR'>adjoint-free</span></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'> optimal control law. For each arc, one of the following cases is possible:</span><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span style='font-size:12.0pt;font-family:"Times New Roman";mso-fareast-font-family: "Times New Roman"'><span style='mso-tab-count:1'>&emsp;&emsp;</span>1. The optimal input </span><i><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-bidi-font-family:"Arial Unicode MS"'>u<sub>j</sub></span></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'> is determined by the active path constraint <i>g<sub>k</sub></i>(<b>x</b>,<b>u</b>) = 0.</span><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span style='font-size:12.0pt;font-family:"Times New Roman";mso-fareast-font-family: "Times New Roman"'><span style='mso-tab-count:1'>&emsp;&emsp;</span>2. </span><span lang=FR style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:FR'>T</span><span style='font-size:12.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>he optimal input <i>u<sub>j</sub></i> is determined by the active path constraint <i>h<sub>k</sub></i>(<b>x</b>) ≤ </span><span lang=RU style='font-size:12.0pt;font-family:"Times New Roman"; mso-bidi-font-family:"Arial Unicode MS";mso-ansi-language:RU'>0, </span><span lang=FR style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:FR'>and </span><i><span style='font-size: 12.0pt;font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>u<sub>j</sub></span></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'> is obtained such that </span><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'> ∂<i>h<sub>k</sub></i>/∂<b>x</b>(<b>x</b>) <b>f</b>(<b>x</b>,<b>u</b>) = 0.</span><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span style='font-size:12.0pt;font-family:"Times New Roman";mso-fareast-font-family: "Times New Roman"'><span style='mso-tab-count:1'>&emsp;&emsp;</span>3. Otherwise, the optimal input </span><i><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-bidi-font-family:"Arial Unicode MS"'>u<sub>j</sub></span></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'> is determined by the condition det(</span><span style='font-size:12.0pt;font-family:"Arial Unicode MS"'>ℳ</span><i><sub><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>u</span></sub></i><i><sub><span style='font-size:10.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"; position:relative;top:2.0pt;mso-text-raise:-2.0pt'>ⱼ</span></sub></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>) = 0, where</span><span style='font-size:12.0pt; font-family:"Times New Roman";mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span style='font-size:12.0pt;font-family:"Times New Roman";mso-fareast-font-family: "Times New Roman"'><span style='mso-tab-count:1'>&emsp;&emsp;</span></span><span style='font-size:12.0pt;font-family:"Arial Unicode MS"'>ℳ</span><i><sub><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>u</span></sub></i><i><sub><span style='font-size:10.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"; position:relative;top:2.0pt;mso-text-raise:-2.0pt'>ⱼ</span></sub></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'><span style="mso-spacerun:yes"> </span>:= [∂<b>f</b><i><sup>u</sup></i></span><i><sup><span style='font-size:10.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";position:relative;top:2.0pt;mso-text-raise:-2.0pt'>ⱼ</span></sup></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>/∂<i>u<sub>j</sub></i>(<b>x</b>,<b>u</b></span><span lang=DE style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:DE'>)<span style="mso-spacerun:yes">&emsp; </span></span><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-bidi-font-family:"Arial Unicode MS"'>Δ<i><sub>j</sub></i> ∂<b>f</b><i><sup>u</sup></i></span><i><sup><span style='font-size:10.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";position:relative;top:2.0pt;mso-text-raise:-2.0pt'>ⱼ</span></sup></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>/∂<i>u<sub>j</sub></i>(<b>x</b>,<b>u</b></span><span lang=DE style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:DE'>)<span style="mso-spacerun:yes">&emsp; </span></span><span style='font-size:12.0pt;font-family:"Arial Unicode MS"'>⋯</span><span lang=DE style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:DE'><span style="mso-spacerun:yes">&emsp; </span>Δ</span><i><sub><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-bidi-font-family:"Arial Unicode MS"'>j</span></sub></i><i><sub><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:FR'> </span></sub></i><i><sup><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>ρ</span></sup></i><i><sup><span style='font-size:10.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"; position:relative;top:2.0pt;mso-text-raise:-2.0pt'>ⱼ</span></sup></i><sup><span lang=RU style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:RU'>-1</span></sup><span style='font-size: 12.0pt;font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'> ∂<b>f</b><i><sup>u</sup></i></span><i><sup><span style='font-size:10.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"; position:relative;top:2.0pt;mso-text-raise:-2.0pt'>ⱼ</span></sup></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>/∂<i>u<sub>j</sub></i>(<b>x</b>,<b>u</b></span><span lang=PT style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:PT'>)],</span><span style='font-size:12.0pt; font-family:"Times New Roman";mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>and the operators Δ<i><sub>j</sub></i>,…, Δ<i><sub>j</sub></i></span><i><sub><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:FR'> </span></sub></i><i><sup><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>ρ</span></sup></i><i><sup><span style='font-size:10.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"; position:relative;top:2.0pt;mso-text-raise:-2.0pt'>ⱼ</span></sup></i><sup><span lang=RU style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:RU'>-1</span></sup><span style='font-size: 12.0pt;font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'> are defined as</span><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span style='font-size:12.0pt;font-family:"Times New Roman";mso-fareast-font-family: "Times New Roman"'><span style='mso-tab-count:1'>&emsp;&emsp;</span>Δ</span><i><sub><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>j</span></sub></i><span style='font-size:12.0pt;font-family: "Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'> <b>v</b></span><span lang=FR style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:FR'> := </span><span style='font-size: 12.0pt;font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>∂<b>v</b>/∂<b>x</b> <b>f</b>(<b>x</b>,<b>u</b>) - ∂<b>f</b><i><sup>u</sup></i></span><i><sup><span style='font-size:10.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";position:relative;top:2.0pt;mso-text-raise:-2.0pt'>ⱼ</span></sup></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>/∂<b>x</b><i><sup>u</sup></i></span><i><sup><span style='font-size:10.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";position:relative;top:2.0pt;mso-text-raise:-2.0pt'>ⱼ</span></sup></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>(<b>x</b>,<b>u</b>) <b>v</b></span><span lang=DE style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:DE'> + </span><b><span style='font-size: 12.0pt;font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>∑</span></b><i><sub><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>n</span></sub></i><sub><span style='font-size:12.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>≥0</span></sub><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'> ∂<b>v</b>/∂<b>u</b><sup>(<i>n</i>)</sup> <b>u</b><sup>(<i>n</i>+1)</sup>,</span><span style='font-size:12.0pt;font-family:"Times New Roman";mso-fareast-font-family: "Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span style='font-size:12.0pt;font-family:"Times New Roman";mso-fareast-font-family: "Times New Roman"'><span style='mso-tab-count:1'>&emsp;&emsp;</span>Δ</span><i><sub><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>j</span></sub></i><i><sub><span style='font-size:12.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"; mso-ansi-language:FR'> </span></sub></i><i><sup><span style='font-size:12.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>l</span></sup></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'> <b>v</b></span><span lang=FR style='font-size:12.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"; mso-ansi-language:FR'> := </span><span style='font-size:12.0pt;font-family: "Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>Δ<i><sub>j</sub></i> (Δ<i><sub>j</sub></i></span><i><sub><span style='font-size:12.0pt;font-family: "Times New Roman";mso-bidi-font-family:"Arial Unicode MS";mso-ansi-language: FR'> </span></sub></i><i><sup><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-bidi-font-family:"Arial Unicode MS"'>l</span></sup></i><sup><span lang=RU style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:RU'>-1</span></sup><span lang=RU style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'> </span><b><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-bidi-font-family:"Arial Unicode MS"'>v</span></b><span style='font-size: 12.0pt;font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>), <span style="mso-spacerun:yes">&emsp;</span>if <i>l</i> = 2,…,</span> <span style='font-size:12.0pt;font-family:"Times New Roman"; mso-bidi-font-family:"Arial Unicode MS";mso-ansi-language:FR'> </span><i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>ρ<sub>j</sub></span></i><span style='font-size:12.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>-1,</span><span style='font-size:12.0pt;font-family:"Times New Roman";mso-fareast-font-family: "Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>for any vector field <b>v</b> of dimension <i>ρ<sub>j</sub></i>, with <b>x</b><i><sup>u</sup></i></span><i><sup><span style='font-size:10.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"; position:relative;top:2.0pt;mso-text-raise:-2.0pt'>ⱼ</span></sup></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'><span style="mso-spacerun:yes"> </span>being the <i>ρ<sub>j</sub></i>-dimensional vector of states that can be influenced by manipulating <i>u<sub>j</sub></i>, and <b>f</b><i><sup>u</sup></i></span><i><sup><span style='font-size:10.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"; position:relative;top:2.0pt;mso-text-raise:-2.0pt'>ⱼ</span></sup></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>(<b>x</b>,<b>u</b>) such that <b>ẋ</b><i><sup>u</sup></i></span><i><sup><span style='font-size:10.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";position:relative;top:2.0pt;mso-text-raise:-2.0pt'>ⱼ</span></sup></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'><span style="mso-spacerun:yes"> </span>= <b>f</b><i><sup>u</sup></i></span><i><sup><span style='font-size:10.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";position:relative;top:2.0pt;mso-text-raise:-2.0pt'>ⱼ</span></sup></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>(<b>x</b>,<b>u</b>). </span><span style='font-size:12.0pt; font-family:"Times New Roman";mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>However, the input <i>u<sub>j</sub></i> and its time derivatives may not appear explicitly in the function det(</span><span style='font-size:12.0pt;font-family:"Arial Unicode MS"'>ℳ</span><i><sub><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>u</span></sub></i><i><sub><span style='font-size:10.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"; position:relative;top:2.0pt;mso-text-raise:-2.0pt'>ⱼ</span></sub></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>). Hence, as a general approach to find the optimal input <i>u<sub>j</sub></i> when it is not determined by an active path constraint, the function det(</span><span style='font-size:12.0pt;font-family:"Arial Unicode MS"'>ℳ</span><i><sub><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>u</span></sub></i><i><sub><span style='font-size:10.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"; position:relative;top:2.0pt;mso-text-raise:-2.0pt'>ⱼ</span></sub></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>) is subject to time differentiation until <i>u<sub>j</sub></i> or one of its time derivatives appears in d<i><sup>r</sup></i></span><i><sup><span style='font-size:10.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";position:relative;top:2.0pt;mso-text-raise:-2.0pt'>ⱼ</span></sup></i><span lang=DE style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:DE'>(det(</span><span style='font-size: 12.0pt;font-family:"Arial Unicode MS"'>ℳ</span><i><sub><span style='font-size: 12.0pt;font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>u</span></sub></i><i><sub><span style='font-size:10.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";position:relative;top:2.0pt;mso-text-raise:-2.0pt'>ⱼ</span></sub></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>))/d<i>t<sup>r</sup></i></span><i><sup><span style='font-size:10.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";position:relative;top:2.0pt;mso-text-raise:-2.0pt'>ⱼ</span></sup></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>, for some <i>r<sub>j</sub></i>. Let <i>u<sub>j</sub></i><sup>(<i>ξ</i></sup></span><i><sup><span style='font-size:10.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";position:relative;top:2.0pt;mso-text-raise:-2.0pt'>ⱼ</span></sup></i><sup><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>)</span></sup><span style='font-size:12.0pt;font-family: "Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'> be the highest-order time derivative of <i>u<sub>j</sub></i> that appears in d<i><sup>r</sup></i></span><i><sup><span style='font-size:10.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";position:relative;top:2.0pt;mso-text-raise:-2.0pt'>ⱼ</span></sup></i><span lang=DE style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:DE'>(det(</span><span style='font-size: 12.0pt;font-family:"Arial Unicode MS"'>ℳ</span><i><sub><span style='font-size: 12.0pt;font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>u</span></sub></i><i><sub><span style='font-size:10.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";position:relative;top:2.0pt;mso-text-raise:-2.0pt'>ⱼ</span></sub></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>))/d<i>t<sup>r</sup></i></span><i><sup><span style='font-size:10.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";position:relative;top:2.0pt;mso-text-raise:-2.0pt'>ⱼ</span></sup></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>. Then, the optimal input <i>u<sub>j</sub></i><sup>(<i>ξ</i></sup></span><i><sup><span style='font-size:10.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";position:relative;top:2.0pt;mso-text-raise:-2.0pt'>ⱼ</span></sup></i><sup><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>)</span></sup><span style='font-size:12.0pt;font-family: "Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'> is obtained such that </span><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'> </span><span lang=FR style='font-size:12.0pt;font-family: "Times New Roman";mso-bidi-font-family:"Arial Unicode MS";mso-ansi-language: FR'>d<i><sup>r</sup></i></span><i><sup><span lang=FR style='font-size:10.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"; position:relative;top:2.0pt;mso-text-raise:-2.0pt;mso-ansi-language:FR'>ⱼ</span></sup></i><span lang=DE style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:DE'>(det(</span><span lang=FR style='font-size:12.0pt;font-family:"Arial Unicode MS";mso-ansi-language:FR'>ℳ</span><i><sub><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>u</span></sub></i><i><sub><span lang=FR style='font-size: 10.0pt;font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"; position:relative;top:2.0pt;mso-text-raise:-2.0pt;mso-ansi-language:FR'>ⱼ</span></sub></i><span lang=FR style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:FR'>)</span><span style='font-size:12.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>)</span><span lang=FR style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:FR'>/d</span><i><span style='font-size: 12.0pt;font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>t</span></i><i><sup><span lang=FR style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:FR'>r</span></sup></i><i><sup><span lang=FR style='font-size:10.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";position:relative;top:2.0pt;mso-text-raise:-2.0pt; mso-ansi-language:FR'>ⱼ</span></sup></i><span lang=RU style='font-size:12.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"; mso-ansi-language:RU'><span style="mso-spacerun:yes"> </span>= 0</span><span lang=FR style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:FR'>.</span><span style='font-size:12.0pt; font-family:"Times New Roman";mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>The mass and heat balances for batch and semi-batch reactors can be written using the concept of extents [2]. Let us consider a homogeneous batch or semi-batch reactor with <i>R</i> independent reactions and <i>p</i> independent inlets (</span><i><span lang=NL style='font-size:12.0pt;font-family: "Times New Roman";mso-bidi-font-family:"Arial Unicode MS";mso-ansi-language: NL'>p </span></i><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-bidi-font-family:"Arial Unicode MS"'>= 0 for batch reactors), where <b>u</b><i><sub>in</sub></i>(<i>t</i>) is the <i>p</i>-dimensional vector of inlet flowrates, and </span><i><span lang=FR style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:FR'>q</span></i><i><sub><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>ex</span></sub></i><span style='font-size:12.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>(<i>t</i>) is the exchanged heat power. The numbers of moles <b>n</b>(<i>t</i>) and the heat </span><i><span lang=FR style='font-size:12.0pt;font-family:"Times New Roman"; mso-bidi-font-family:"Arial Unicode MS";mso-ansi-language:FR'>Q</span></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>(<i>t</i>) can be expressed as a linear combination of extents, according to</span><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><b><span style='font-size:12.0pt;font-family:"Times New Roman";mso-fareast-font-family: "Times New Roman"'><span style='mso-tab-count:1'>&emsp;&emsp;</span>n</span></b><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>(<i>t</i>) = <b>N</b><sup>T</sup> <b>x</b><i><sub>r</sub></i>(<i>t</i>) + <b>W</b><i><sub>in</sub></i> <b>x</b><i><sub>in</sub></i>(<i>t</i>) + <b>n</b><sub>0</sub>,</span><span style='font-size:12.0pt;font-family:"Times New Roman";mso-fareast-font-family: "Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><i><span lang=FR style='font-size:12.0pt;font-family:"Times New Roman";mso-fareast-font-family: "Times New Roman";mso-ansi-language:FR'><span style='mso-tab-count:1'>&emsp;&emsp;</span>Q</span></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>(<i>t</i>) = (-Δ<b>H</b>)<sup>T</sup> <b>x</b><i><sub>r</sub></i>(<i>t</i>) + <b>Ť</b><i><sub>in</sub></i><sup>T</sup><i> </i><b>x</b><i><sub>in</sub></i>(<i>t</i>) + <i>x<sub>ex</sub></i>(<i>t</i>) + </span><i><span lang=FR style='font-size: 12.0pt;font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"; mso-ansi-language:FR'>Q</span></i><sub><span style='font-size:12.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>0</span></sub><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>,</span><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>where <b>n</b><sub>0</sub> is the <i>S</i>-dimensional vector of initial numbers of moles, </span><i><span lang=FR style='font-size: 12.0pt;font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"; mso-ansi-language:FR'>Q</span></i><sub><span style='font-size:12.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>0</span></sub><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'> is the initial heat, <b>N</b> is the <i>R</i>×<i>S</i></span><span lang=IT style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:IT'> stoichiometric matrix, </span><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>Δ<b>H</b> is the <i>R</i>-dimensional vector of heats of reaction, <b>W</b><i><sub>in</sub></i> is the <i>S</i>×<i>p</i></span><span lang=FR style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:FR'> inlet-composition matrix, </span><b><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>Ť</span></b><i><sub><span style='font-size:12.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>in</span></sub></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'> is the <i>p</i>-dimensional vector of inlet specific enthalpies, <b>x</b><i><sub>r</sub></i>(<i>t</i>) is the <i>R</i>-dimensional vector of extents of reaction, <b>x</b><i><sub>in</sub></i>(<i>t</i>) is the <i>p</i>-dimensional vector of extents of inlet, and <i>x<sub>ex</sub></i>(<i>t</i>) is the extent of heat exchange.</span><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>The state vector of dimension <i>n<sub>x</sub></i></span><span lang=FR style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:FR'> := </span><i><span style='font-size: 12.0pt;font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>R</span></i><span lang=DE style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:DE'> + </span><i><span style='font-size: 12.0pt;font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>p</span></i><span lang=DE style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:DE'> + 1 is</span><span style='font-size: 12.0pt;font-family:"Times New Roman";mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><b><span style='font-size:12.0pt;font-family:"Times New Roman";mso-fareast-font-family: "Times New Roman"'><span style='mso-tab-count:1'>&emsp;&emsp;</span>x</span></b><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>(<i>t</i>) := [<b>x</b><i><sub>r</sub></i>(<i>t</i>)<sup>T</sup></span><span lang=DE style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:DE'><span style="mso-spacerun:yes">&emsp; </span></span><b><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-bidi-font-family:"Arial Unicode MS"'>x</span></b><i><sub><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>in</span></sub></i><span style='font-size:12.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>(<i>t</i>)<sup>T</sup></span><span lang=DE style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:DE'><span style="mso-spacerun:yes">&emsp; </span></span><i><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-bidi-font-family:"Arial Unicode MS"'>x<sub>ex</sub></span></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>(<i>t</i></span><span lang=PT style='font-size:12.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"; mso-ansi-language:PT'>)]</span><sup><span style='font-size:12.0pt;font-family: "Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>T</span></sup><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>,</span><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>while the input vector of dimension <i>n<sub>u</sub></i></span><span lang=FR style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:FR'> := </span><i><span style='font-size: 12.0pt;font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>p</span></i><span lang=DE style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:DE'> + 1 is</span><span style='font-size: 12.0pt;font-family:"Times New Roman";mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><b><span style='font-size:12.0pt;font-family:"Times New Roman";mso-fareast-font-family: "Times New Roman"'><span style='mso-tab-count:1'>&emsp;&emsp;</span>u</span></b><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>(<i>t</i>) := [<b>u</b><i><sub>in</sub></i>(<i>t</i>)<sup>T</sup></span><span lang=DE style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:DE'><span style="mso-spacerun:yes">&emsp; </span></span><i><span lang=FR style='font-size:12.0pt;font-family:"Times New Roman"; mso-bidi-font-family:"Arial Unicode MS";mso-ansi-language:FR'>q</span></i><i><sub><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>ex</span></sub></i><span style='font-size:12.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>(<i>t</i></span><span lang=PT style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:PT'>)]</span><sup><span style='font-size: 12.0pt;font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>T</span></sup><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>.</span><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>The dynamic equations can be written compactly as</span><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span style='font-size:12.0pt;font-family:"Times New Roman";mso-fareast-font-family: "Times New Roman"'><span style='mso-tab-count:1'>&emsp;&emsp;</span><b>ẋ</b></span><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-bidi-font-family:"Arial Unicode MS"'>(<i>t</i>) = <b>f</b>(<b>x</b>(<i>t</i>),<b>u</b>(<i>t</i>)),<span style="mso-spacerun:yes">&emsp;</span><b>x</b>(<i>t</i><sub>0</sub>) = <b>0</b><i><sub>R</sub></i><sub>+<i>p</i>+1</sub>,</span><span style='font-size:12.0pt;font-family:"Times New Roman";mso-fareast-font-family: "Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><b><span style='font-size:12.0pt;font-family:"Times New Roman";mso-fareast-font-family: "Times New Roman"'><span style='mso-tab-count:1'>&emsp;&emsp;</span>f</span></b><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>(<b>x</b>(<i>t</i>),<b>u</b>(<i>t</i>)) := [<b>r</b><i><sub>v</sub></i>(<b>x</b>(<i>t</i>))<sup>T</sup></span><span lang=DE style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:DE'><span style="mso-spacerun:yes">&emsp; </span></span><b><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-bidi-font-family:"Arial Unicode MS"'>u</span></b><span style='font-size: 12.0pt;font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>(<i>t</i>)<sup>T</sup></span><span lang=PT style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:PT'>]</span><sup><span style='font-size: 12.0pt;font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>T</span></sup><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>,</span><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>where <b>r</b><i><sub>v</sub></i>(<b>x</b>(<i>t</i>)) is the <i>R</i>-dimensional vector of reaction rates. In batch and semi-batch reactors, with <i>ẋ<sub>j</sub></i> = <i>f<sub>j</sub></i>(<b>x</b>,<b>u</b></span><span lang=FR style='font-size: 12.0pt;font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"; mso-ansi-language:FR'>) := </span><i><span style='font-size:12.0pt;font-family: "Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>u<sub>j</sub></span></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>, it is possible to define the following vectors of dimension <i>ρ<sub>j</sub></i></span><span lang=FR style='font-size:12.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"; mso-ansi-language:FR'> := </span><i><span style='font-size:12.0pt;font-family: "Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>R</span></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>+1:</span><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><b><span style='font-size:12.0pt;font-family:"Times New Roman";mso-fareast-font-family: "Times New Roman"'><span style='mso-tab-count:1'>&emsp;&emsp;</span>x</span></b><i><sup><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>u</span></sup></i><i><sup><span style='font-size:10.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"; position:relative;top:2.0pt;mso-text-raise:-2.0pt'>ⱼ</span></sup></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'><span style="mso-spacerun:yes"> </span>:= [<b>x</b><i><sub>r</sub></i><sup>T</sup></span><span lang=DE style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:DE'><span style="mso-spacerun:yes">&emsp; </span></span><i><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-bidi-font-family:"Arial Unicode MS"'>x<sub>j</sub></span></i><span lang=PT style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:PT'>]</span><sup><span style='font-size: 12.0pt;font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>T</span></sup><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>,</span><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><b><span style='font-size:12.0pt;font-family:"Times New Roman";mso-fareast-font-family: "Times New Roman"'><span style='mso-tab-count:1'>&emsp;&emsp;</span>f</span></b><i><sup><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>u</span></sup></i><i><sup><span style='font-size:10.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"; position:relative;top:2.0pt;mso-text-raise:-2.0pt'>ⱼ</span></sup></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>(<b>x</b>,<b>u</b>) := [<b>r</b><i><sub>v</sub></i>(<b>x</b>)<sup>T</sup></span><span lang=DE style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:DE'><span style="mso-spacerun:yes">&emsp; </span></span><i><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-bidi-font-family:"Arial Unicode MS"'>u<sub>j</sub></span></i><span lang=PT style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:PT'>]</span><sup><span style='font-size: 12.0pt;font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>T</span></sup><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>.</span><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>Note that, since this system is input-affine, det(</span><span style='font-size:12.0pt;font-family:"Arial Unicode MS"'>ℳ</span><i><sub><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>u</span></sub></i><i><sub><span style='font-size:10.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"; position:relative;top:2.0pt;mso-text-raise:-2.0pt'>ⱼ</span></sub></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>) and its time derivatives are polynomial functions of <i>u<sub>j</sub></i> and its time derivatives, thus resulting in a finite number of solutions, and typically a single solution that satisfies the condition in Case 3.</span><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>Let us define the state vector <b>x̌</b><i><sub>j</sub></i> as the complement of the state <i>x<sub>j</sub></i> (all states <b>x</b> except <i>x<sub>j</sub></i>), and the vector <b>f̌</b><i><sub>j</sub></i>(<b>x</b>,<b>u</b>) as the corresponding complement of <i>f<sub>j</sub></i>(<b>x</b>,<b>u</b>). Then, one can prove that, when the optimal input <i>u<sub>j</sub></i> is not determined by an active path constraint:</span><span style='font-size:12.0pt; font-family:"Times New Roman";mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span style='font-size:12.0pt;font-family:"Times New Roman";mso-fareast-font-family: "Times New Roman"'><span style='mso-tab-count:1'>&emsp;&emsp;</span></span><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>1. For reactors with a single independent reaction, the input <i>u<sub>j</sub></i> is determined by</span><span style='font-size:12.0pt; font-family:"Times New Roman";mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span lang=DE style='font-size:12.0pt;font-family:"Times New Roman";mso-fareast-font-family: "Times New Roman";mso-ansi-language:DE'><span style='mso-tab-count:1'>&emsp;&emsp;</span>d(det(</span><span style='font-size:12.0pt;font-family:"Arial Unicode MS"'>ℳ</span><i><sub><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>u</span></sub></i><i><sub><span style='font-size:10.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"; position:relative;top:2.0pt;mso-text-raise:-2.0pt'>ⱼ</span></sub></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>))/d<i>t</i> = ∂(∂<b>r</b><i><sub>v</sub></i>/∂<i>x<sub>j</sub></i>(<b>x</b>))/∂<i>x<sub>j</sub></i> <i>u<sub>j</sub></i></span><span lang=FR style='font-size:12.0pt;font-family: "Times New Roman";mso-bidi-font-family:"Arial Unicode MS";mso-ansi-language: FR'> + </span><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-bidi-font-family:"Arial Unicode MS"'>∂(∂<b>r</b><i><sub>v</sub></i>/∂<i>x<sub>j</sub></i>(<b>x</b>))/∂<b>x̌</b><i><sub>j</sub></i> <b>f̌</b><i><sub>j</sub></i>(<b>x</b>,<b>u</b>),</span><span style='font-size: 12.0pt;font-family:"Times New Roman";mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>since <i>u<sub>j</sub></i> and its time derivatives do not appear in</span><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span style='font-size:12.0pt;font-family:"Times New Roman";mso-fareast-font-family: "Times New Roman"'><span style='mso-tab-count:1'>&emsp;&emsp;</span>det(</span><span style='font-size:12.0pt;font-family:"Arial Unicode MS"'>ℳ</span><i><sub><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>u</span></sub></i><i><sub><span style='font-size:10.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"; position:relative;top:2.0pt;mso-text-raise:-2.0pt'>ⱼ</span></sub></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>) = ∂<b>r</b><i><sub>v</sub></i>/∂<i>x<sub>j</sub></i>(<b>x</b>).</span><span style='font-size:12.0pt;font-family:"Times New Roman";mso-fareast-font-family: "Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span style='font-size:12.0pt;font-family:"Times New Roman";mso-fareast-font-family: "Times New Roman"'><span style='mso-tab-count:1'>&emsp;&emsp;</span></span><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>2. For reactors with 2 independent reactions, the input <i>u<sub>j</sub></i> is determined by</span><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span style='font-size:12.0pt;font-family:"Times New Roman";mso-fareast-font-family: "Times New Roman"'><span style='mso-tab-count:1'>&emsp;&emsp;</span>det(</span><span style='font-size:12.0pt;font-family:"Arial Unicode MS"'>ℳ</span><i><sub><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>u</span></sub></i><i><sub><span style='font-size:10.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"; position:relative;top:2.0pt;mso-text-raise:-2.0pt'>ⱼ</span></sub></i><span lang=PT style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:PT'>) = det([</span><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>∂<b>r</b><i><sub>v</sub></i>/∂<i>x<sub>j</sub></i>(<b>x</b></span><span lang=DE style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:DE'>)<span style="mso-spacerun:yes">&emsp; </span></span><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-bidi-font-family:"Arial Unicode MS"'>∂(∂<b>r</b><i><sub>v</sub></i>/∂<i>x<sub>j</sub></i>(<b>x</b>))/∂<i>x<sub>j</sub></i></span><span lang=PT style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:PT'>]) </span><i><span style='font-size: 12.0pt;font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>u<sub>j</sub></span></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-fareast-font-family: "Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span lang=FR style='font-size:12.0pt;font-family:"Times New Roman";mso-fareast-font-family: "Times New Roman";mso-ansi-language:FR'><span style='mso-tab-count:1'>&emsp;&emsp;</span><span style="mso-spacerun:yes">&emsp;</span>+</span><span lang=PT style='font-size: 12.0pt;font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"; mso-ansi-language:PT'> det([</span><span style='font-size:12.0pt;font-family: "Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'>∂<b>r</b><i><sub>v</sub></i>/∂<i>x<sub>j</sub></i>(<b>x</b></span><span lang=DE style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:DE'>)<span style="mso-spacerun:yes">&emsp; </span>-</span><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-bidi-font-family:"Arial Unicode MS"'>∂<b>r</b><i><sub>v</sub></i>/∂<b>x</b><i><sub>r</sub></i>(<b>x</b>) ∂<b>r</b><i><sub>v</sub></i>/∂<i>x<sub>j</sub></i>(<b>x</b>) + ∂(∂<b>r</b><i><sub>v</sub></i>/∂<i>x<sub>j</sub></i>(<b>x</b>))/∂<b>x̌</b><i><sub>j</sub></i> <b>f̌</b><i><sub>j</sub></i>(<b>x</b>,<b>u</b></span><span lang=PT style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:PT'>)])</span><span style='font-size:12.0pt; font-family:"Times New Roman";mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>One can use symbolic computation software to evaluate the function det(</span><span style='font-size:12.0pt;font-family:"Arial Unicode MS"'>ℳ</span><i><sub><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>u</span></sub></i><i><sub><span style='font-size:10.0pt; font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"; position:relative;top:2.0pt;mso-text-raise:-2.0pt'>ⱼ</span></sub></i><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>) and its time derivatives and obtain the optimal input <i>u<sub>j</sub></i> or one of its time derivatives when it is not determined by an active path constraint.</span><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>The advantage of the proposed approach is that it reduces the set of possible arcs to a finite number of possibilities. This, in turn, results in a finite number of arc sequences if one assumes an upper bound on the number of arcs present in the optimal solution. Hence, instead of solving the original infinite-dimensional problem, one can simply perform numerical optimization for each arc sequence, using the switching times between arcs and the initial conditions as decision variables.</span><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>Note that the dynamic state equations can be integrated forward in time, since it is possible to evaluate the corresponding inputs without knowledge of the adjoint variables. Once the forward integration is complete, one can integrate backward in time to obtain the corresponding adjoint variables, which enables the computation of the gradients with respect to the switching times and initial conditions of the arcs.</span><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span lang=FR style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:FR'>The proposed approach will be illustrated to maximize the final </span><span lang=FR style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:FR'>quantity</span><span style='font-size: 12.0pt;font-family:"Times New Roman";mso-bidi-font-family:"Arial Unicode MS"'> of product in an acetoacetylation reaction</span><span lang=FR style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:FR'>, subject to constraints on the final concentration of reactants and by-products [3]</span><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>.</span><span style='font-size:12.0pt;font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>[1] B. Srinivasan, S. Palanki, and D. Bonvin. Dynamic optimization of batch processes: I. Characterization of the nominal solution, <i>Comp. Chem. Eng.</i>, 27:1-26, 2003.</span><span style='font-size:12.0pt;font-family: "Times New Roman";mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS"'>[2] D. Rodrigues, S. Srinivasan, J. Billeter, and D. Bonvin. Variant and invariant states for chemical reaction systems, <i>Comp. Chem. Eng.</i>, 73:23-33, 2015.</span><span style='font-size:12.0pt;font-family: "Times New Roman";mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <p class=Default style='margin-right:53.0pt;text-align:justify;text-justify: inter-ideograph;tab-stops:28.0pt 56.0pt 84.0pt 112.0pt 140.0pt 168.0pt 196.0pt 224.0pt 252.0pt 280.0pt 308.0pt 336.0pt'><span lang=FR style='font-size:12.0pt;font-family:"Times New Roman";mso-bidi-font-family: "Arial Unicode MS";mso-ansi-language:FR'>[3] B. Chachuat, B. Srinivasan, and D. Bonvin. Adaptation strategies for real-time optimization, <i>Comp. Chem. Eng.</i>, 33:1557-1567, 2009.</span></p> </div> </body> </html>


Presented at:
109th Annual Meeting of the American Institute of Chemical Engineers (AIChE), Minneapolis, Minnesota (USA), October 29 - November 3, 2017
Year:
2017
Laboratories:




 Record created 2017-05-16, last modified 2018-09-13

Abstract:
Download fulltext
PDF

Rate this document:

Rate this document:
1
2
3
 
(Not yet reviewed)