DAD SYSTEMS OF CONTROL AND OBSERVATION AND OPEN PROBLEM

00:00 05-08-2009
<p class="MsoNormal" dir="ltr" style="DIRECTION: ltr; unicode-bidi: embed; TEXT-ALIGN: left"><font face="Arial" size="2"><span style="FONT-SIZE: 14px; FONT-FAMILY: Arial">Abstract. </span></font></p> <p class="MsoNormal" dir="ltr" style="DIRECTION: ltr; unicode-bidi: embed; TEXT-ALIGN: left"><font face="Times New Roman" size="3"><span style="FONT-SIZE: 12pt"></span></font> </p> <p class="MsoNormal" dir="ltr" style="DIRECTION: ltr; LINE-HEIGHT: 150%; unicode-bidi: embed; TEXT-ALIGN: justify"><font face="Arial" size="2"><span style="FONT-SIZE: 14px; LINE-HEIGHT: 150%; FONT-FAMILY: Arial">We consider several kinds of linear hybrid systems described by discrete-difference and difference-differential equations with control.</span></font></p> <p class="MsoNormal" dir="ltr" style="DIRECTION: ltr; LINE-HEIGHT: 150%; unicode-bidi: embed; TEXT-ALIGN: justify"><font face="Arial" size="2"><span style="FONT-SIZE: 14px; LINE-HEIGHT: 150%; FONT-FAMILY: Arial">Special attention is paid to the hybrid difference-differential (differential-algebraic time-delay (DAD)) systems   in symmetric form. </span></font></p> <p class="MsoNormal" dir="ltr" style="DIRECTION: ltr; LINE-HEIGHT: 150%; unicode-bidi: embed; TEXT-ALIGN: justify"><font face="Arial" size="2"><span style="FONT-SIZE: 14px; LINE-HEIGHT: 150%; FONT-FAMILY: Arial">For solutions of such systems, a variation-of-constants formula is proposed and the relative controllability-observability principle is established. </span></font></p> <p class="MsoNormal" dir="ltr" style="DIRECTION: ltr; LINE-HEIGHT: 150%; unicode-bidi: embed; TEXT-ALIGN: justify"><font face="Arial" size="2"><span style="FONT-SIZE: 14px; LINE-HEIGHT: 150%; FONT-FAMILY: Arial">For stationary systems, we introduce the determining equations and give solution representations into series their determining equation solutions. </span></font></p> <p class="MsoNormal" dir="ltr" style="DIRECTION: ltr; LINE-HEIGHT: 150%; unicode-bidi: embed; TEXT-ALIGN: justify"><font face="Arial" size="2"><span style="FONT-SIZE: 14px; LINE-HEIGHT: 150%; FONT-FAMILY: Arial">Then algebraic properties of the determining equation solutions are investigated, in particular, the well-known Hamilton-Cayley matrix theorem </span></font></p> <p class="MsoNormal" dir="ltr" style="DIRECTION: ltr; LINE-HEIGHT: 150%; unicode-bidi: embed; TEXT-ALIGN: justify"><font face="Arial" size="2"><span style="FONT-SIZE: 14px; LINE-HEIGHT: 150%; FONT-FAMILY: Arial">is extended to the solutions of the system determining equations. </span></font></p> <p class="MsoNormal" dir="ltr" style="DIRECTION: ltr; LINE-HEIGHT: 150%; unicode-bidi: embed; TEXT-ALIGN: justify"><font face="Arial" size="2"><span style="FONT-SIZE: 14px; LINE-HEIGHT: 150%; FONT-FAMILY: Arial">As a result, parametric criteria for the relative controllability and relative observability are given. Some open problems are formulated. <br /></span></font> </p> <p class="MsoNormal" dir="ltr" style="DIRECTION: ltr; unicode-bidi: embed; TEXT-ALIGN: left"><font face="Arial" size="2"><span style="FONT-SIZE: 14px; FONT-FAMILY: Arial"></span></font></p> <p class="MsoNormal" dir="ltr" style="DIRECTION: ltr; unicode-bidi: embed; TEXT-ALIGN: left"><font face="Arial" size="2"><span style="FONT-SIZE: 14px; FONT-FAMILY: Arial">Light refreshments will be served at 10:45 am. Everybody is welcome.</span></font></p><SPAN STYLE="display: none"> <P>Acetylenide minutes, pointedly? 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