<?xml version="1.0" encoding="UTF-8"?><xml><records><record><source-app name="Biblio" version="7.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">R. HILSCHER</style></author><author><style face="normal" font="default" size="100%">V. ZEIDAN</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">LEGENDRE, JACOBI, AND RICCATI TYPE CONDITIONS FOR TIME SCALE VARIATIONAL PROBLEM WITH APPLICATION</style></title><secondary-title><style face="normal" font="default" size="100%">Dynamic Systems and Applications</style></secondary-title></titles><keywords><keyword><style  face="normal" font="default" size="100%">39A12</style></keyword><keyword><style  face="normal" font="default" size="100%">49K99</style></keyword></keywords><dates><year><style  face="normal" font="default" size="100%">2007</style></year><pub-dates><date><style  face="normal" font="default" size="100%">2007</style></date></pub-dates></dates><urls><web-urls><url><style face="normal" font="default" size="100%">https://acadsol.eu/dsa/articles/16/451-480-Hilscher.pdf</style></url></web-urls></urls><volume><style face="normal" font="default" size="100%">16</style></volume><pages><style face="normal" font="default" size="100%">30</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;p&gt;A time scale quadratic problem J with piecewise right-dense continuous coefficients and one varying endpoint is considered. Such problems are “hybrid”, since they include mixing of continuous- and discrete-time problems. A new notion of a generalized conjugate point involving “dynamic” (hybrid) systems and comprising as special cases those known for the continuous- and discrete-time settings is introduced. A type of a strengthened Legendre condition is identified and used to establish characterizations of the nonnegativity and positivity of J in terms of (i) the nonexistence of such conjugate points, (ii) the natural conjoined basis of the associated time scale Jacobi equation, and (iii) a solution of the corresponding time scale Riccati equation. These results furnish second order necessary optimality conditions for a nonlinear time scale variational problem. Furthermore, we present an example of an optimal impulsive control problem and we show how this problem can be reduced to a variational problem over a time scale.&lt;/p&gt;
</style></abstract><section><style face="normal" font="default" size="100%">451</style></section></record></records></xml>