Linear‐quadratic control problems with L1‐control cost. (27th June 2014)

• Record Type: Journal Article
• Authors: Alt, Walter; Schneider, Christopher
• Is Part Of: Optimal control applications and methods. Volume 36:Number 4(2015:Jul./Aug.)
• Page Start: 512
• Page End: 534
• Publication Date: 2014-06-27
• Subjects: Control theory -- Periodicals; Mathematical optimization -- Periodicals; 629.8312
• DOI: 10.1002/oca.2126
• Languages: English
• ISSNs: 0143-2087

Abstract

<abstract abstract-type="main" id="oca2126-abs-0001"> <title>Summary</title> <p id="oca2126-para-0001">We analyze a class of linear‐quadratic optimal control problems with an additional <italic>L</italic><sup>1</sup>‐control cost depending on a parameter <italic>β</italic>. To deal with this nonsmooth problem, we use an augmentation approach known from linear programming in which the number of control variables is doubled. It is shown that if the optimal control for a given <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj22ddts5n" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:1432087:media:oca2126:oca2126-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mo class="MathClass-bin">*</mml:mo></mml:mrow></mml:msup><mml:mi>⩾</mml:mi><mml:mn>0</mml:mn></mml:math></alternatives></inline-formula> is bang‐zero‐bang and the switching function has a stable structure, the solutions are Lipschitz continuous functions of the parameter <italic>β</italic>. We also show that in this case the optimal controls for <italic>β</italic><sup> * </sup> and a <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj22ddts43" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:1432087:media:oca2126:oca2126-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>β ⩾</mml:mi><mml:mn>0</mml:mn></mml:math></alternatives></inline-formula> with | <italic>β</italic> − <italic>β</italic><sup> * </sup> | sufficiently small coincide except on a set of measure <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj22ddts3j" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:1432087:media:oca2126:oca2126-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">O</mml:mi><mml:mrow><mml:mo class="MathClass-open">(</mml:mo><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mo class="MathClass-close">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>. Finally, we use the augmentation approach to derive error estimates for Euler discretizations. Copyright © 2014 John Wiley & Sons, Ltd.</p> </abstract>