ÏßÐÔϵͳµÄÎȶ¨ÐÔ·ÖÎö - ͼÎÄ ÏÂÔØ±¾ÎÄ

ͼ3.2 £¨a£©Îȶ¨Æ½ºâ״̬¼°Ò»ÌõµäÐ͹켣

£¨b£©½¥½üÎȶ¨Æ½ºâ״̬¼°Ò»ÌõµäÐ͹켣 £¨c£©²»Îȶ¨Æ½ºâ״̬¼°Ò»ÌõµäÐ͹켣

ͼ3.2£¨a£©¡¢(b)ºÍ(c)·Ö±ð±íʾƽºâ״̬¼°¶ÔÓ¦ÓÚÎȶ¨ÐÔ¡¢½¥½üÎȶ¨ÐԺͲ»Îȶ¨ÐԵĵäÐ͹켣¡£ÔÚͼ3.2(a)¡¢(b)ºÍ(c)ÖУ¬ÓòS£¨?£©ÖÆÔ¼×ųõʼ״̬x0£¬¶øÓòS(?£©ÊÇÆðʼÓÚx0µÄ¹ì¼£µÄ±ß½ç¡£

×¢Ò⣬ÓÉÓÚÉÏÊö¶¨Òå²»ÄÜÏêϸµØËµÃ÷¿ÉÈÝÐí³õʼÌõ¼þµÄ¾«È·ÎüÒýÓò£¬Òò¶ø³ý·ÇS(?)¶ÔÓ¦ÓÚÕû¸ö×´Ì¬Æ½Ãæ£¬·ñÔòÕâЩ¶¨ÒåÖ»ÄÜÓ¦ÓÃÓÚÆ½ºâ״̬µÄÁÚÓò¡£

´ËÍ⣬ÔÚͼ5.2£¨c£©ÖУ¬¹ì¼£À뿪ÁËS(?£©£¬Õâ˵Ã÷ƽºâ״̬ÊDz»Îȶ¨µÄ¡£È»¶øÈ´²»ÄÜ˵Ã÷¹ì¼£½«Ç÷ÓÚÎÞÇîÔ¶´¦£¬ÕâÊÇÒòΪ¹ì¼£»¹¿ÉÄÜÇ÷ÓÚÔÚS(?£©ÍâµÄij¸ö¼«ÏÞ»·£¨Èç¹ûÏßÐÔ¶¨³£ÏµÍ³ÊDz»Îȶ¨µÄ£¬ÔòÔÚ²»Îȶ¨Æ½ºâ״̬¸½½ü³ö·¢µÄ¹ì¼£½«Ç÷ÓÚÎÞÇîÔ¶¡£µ«ÔÚ·ÇÏßÐÔϵͳÖУ¬ÕâÒ»½áÂÛ²¢²»Ò»¶¨ÕýÈ·£©¡£

ÉÏÊö¸÷¶¨ÒåµÄÄÚÈÝ£¬¶ÔÓÚÀí½â±¾Õ½éÉܵÄÏßÐԺͷÇÏßÐÔϵͳµÄÎȶ¨ÐÔ·ÖÎö£¬ÊÇ×îµÍÏ޶ȵÄÒªÇó¡£×¢Ò⣬ÕâЩ¶¨Òå²»ÊÇÈ·¶¨Æ½ºâ״̬Îȶ¨ÐÔ¸ÅÄîµÄΨһ·½·¨¡£Êµ¼ÊÉÏ£¬ÔÚÆäËûÎÄÏ×Öл¹ÓÐÁíÍâµÄ¶¨Òå¡£

¶ÔÓÚÏßÐÔϵͳ£¬½¥½üÎȶ¨µÈ¼ÛÓÚ´ó·¶Î§½¥½üÎȶ¨¡£µ«¶ÔÓÚ·ÇÏßÐÔϵͳ£¬Ò»°ãÖ»¿¼ÂÇÎüÒýÇøÎªÓÐÏ޵͍·¶Î§µÄ½¥½üÎȶ¨¡£

×îºóÖ¸³ö£¬ÔÚ¾­µä¿ØÖÆÀíÂÛÖУ¬ÎÒÃÇÒѾ­Ñ§¹ýÎȶ¨ÐÔ¸ÅÄËüÓëLyapunovÒâÒåϵÄÎȶ¨ÐÔ¸ÅÄîÊÇÓÐÒ»¶¨µÄÇø±ðµÄ£¬ÀýÈ磬ÔÚ¾­µä¿ØÖÆÀíÂÛÖÐÖ»Óн¥½üÎȶ¨µÄϵͳ²Å³ÆÎªÎȶ¨µÄϵͳ¡£ÔÚLyapunovÒâÒåÏÂÊÇÎȶ¨µÄ£¬µ«È´²»Êǽ¥½üÎȶ¨µÄϵͳ£¬Ôò½Ð×ö²»Îȶ¨ÏµÍ³¡£Á½ÕßµÄÇø±ðÓëÁªÏµÈçϱíËùʾ¡£

¾­µä¿ØÖÆÀíÂÛ(ÏßÐÔϵͳ£© LyapunovÒâÒåÏ ²»Îȶ¨ (Re(s)>0) ²»Îȶ¨ ÁÙ½çÇé¿ö (Re(s)=0) Îȶ¨ Îȶ¨ (Re(s)<0) ½¥½üÎȶ¨ 3.3.2 ÀîÑÅÆÕŵ·òµÚÒ»·¨

ÀîÑÅÆÕŵ·òµÚÒ»·¨ÊÇͨ¹ýϵͳ¾ØÕóAµÄÌØÕ÷ÖµÀ´ÅжÏϵͳµÄÎȶ¨ÐԵ쬯äÖ÷ÒªÄÚÈÝ: (1)ÓÃÒ»´Î½üËÆ±í´ïʽ±í´ï״̬·½³Ì£¬¼´X?AX£¬¼ÙÈçϵͳ¾ØÕóAdeÈ«²¿ÌØÕ÷Öµ¾ßÓиºÊµ²¿£¬ÔòϵͳÔÚÆ½ºâµã´¦ÊÇÎȶ¨µÄ£¬¶øÇÒÎȶ¨ÐÔÓë¸ß½×µ¼ÊýÎ޹ء£

£¨2£©Èç¹ûÔÚÒ»´Î½üËÆÊ½µÄϵͳ¾ØÕóAµÄÌØÕ÷ÖµÖÐÖÁÉÙÓÐÒ»¸ö¾ßÓÐÕýʵ²¿Ê±£¬ÎÞÂ۸߽׵¼ÊýµÄÇé¿öÈçºÎ£¬ÏµÍ³ÔÚÆ½ºâµã´¦²»Îȶ¨¡£

£¨3£©Èç¹ûÔÚÒ»´Î½üËÆÊ½µÄϵͳ¾ØÕóAµÄÌØÕ÷ÖµÖÐÓÐÁãÌØÕ÷Öµ£¬ÏµÍ³µÄÎȶ¨ÐÔÒªÓи߽׵¼Êý¾ö¶¨¡£µ±¸ß½×µ¼ÊýΪÁãʱ£¬ÏµÍ³´¦ÓÚÁÙ½çÎȶ¨×´Ì¬¡£

.3.3.3 ±êÁ¿º¯ÊýµÄÕý¶¨ÐÔ¶¨Òå

Õý¶¨ÐÔ£º±êÁ¿º¯ÊýV(x)ÔÚÓòSÖжÔËùÓзÇÁã״̬(x?0)ÓÐV(x)?0ÇÒV(0)?0£¬³Æ

2ÊÇÕý¶¨µÄ¡£ V(x)ÔÚÓòSÄÚÕý¶¨¡£ÈçV(x)?x12?x2¸º¶¨ÐÔ£º±êÁ¿º¯ÊýV(x)ÔÚÓòSÖжÔËùÓзÇÁãxÓÐV(x)?0ÇÒV(0)?0£¬³ÆV(x)ÔÚÓòSÄÚ¸º¶¨¡£ÈçV(x)??(x1?x2)ÊǸº¶¨µÄ¡£Èç¹ûV(x)ÊǸº¶¨µÄ£¬-V(x)ÔòÒ»¶¨ÊÇÕý¶¨

µÄ¡£

¸º£¨Õý£©°ë¶¨ÐÔ£ºV(0)?0£¬ÇÒV(x)ÔÚÓòSÄÚijЩ״̬´¦ÓÐV(x)?0£¬¶øÆäËü״̬´¦¾ùÓÐV(x)?0£¨V(x)?0£©£¬Ôò³ÆV(x)ÔÚÓòSÄÚ¸º£¨Õý£©°ë¶¨¡£ÉèV(x)Ϊ¸º°ë¶¨£¬Ôò

22?V(x)ΪÕý°ë¶¨¡£ÈçV(x)??(x1?2x2)2ΪÕý°ë¶¨¡£

²»¶¨ÐÔ£ºV(x)ÔÚÓòSÄÚ¿ÉÕý¿É¸º£¬Ôò³ÆV(x)²»¶¨¡£ÈçV(x)?x1x2ÊDz»¶¨µÄ¡£ ¹ØÓÚV(x,t)Õý¶¨ÐÔµÄÌá·¨ÊÇ£º±êÁ¿º¯ÊýV(x,t)ÔÚÓòSÖУ¬¶ÔÓÚt?t0¼°ËùÓзÇÁã״̬ÓÐV(x,t)?0,ÇÒV(0,t)?0£¬Ôò³ÆV(x,t)ÔÚÓòSÄÚÕý¶¨¡£V(x,t)µÄÆäËü¶¨ºÅÐÔÌá·¨Ààͬ¡£

¶þ´ÎÐͺ¯ÊýÊÇÒ»ÀàÖØÒªµÄ±êÁ¿º¯Êý£¬¼Ç

V(x)?xTPx??x1?p11?p1n??x1????? £¨1£©

?xn??????????pn1?pnn????xn??ÆäÖУ¬PΪ¶Ô³Æ¾ØÕó£¬ÓÐpij?pji¡£ÏÔÈ»Âú×ãV(x)?0£¬Æä¶¨ºÅÐÔÓÉÈü¶ûÎ¬Ë¹ÌØ×¼ÔòÅж¨¡£µ±PµÄ¸÷˳ÐòÖ÷×ÓÐÐÁÐʽ¾ù´óÓÚÁãʱ£¬¼´

p11?0,p11p21p12?0,p22p11,pn1p11,(?1)npn1p1n?0 £¨2£© pnnp1n?0 £¨3£© pnnPΪÕý¶¨¾ØÕó£¬ÔòV(x)Õý¶¨¡£µ±PµÄ¸÷˳ÐòÖ÷×ÓÐÐÁÐʽ¸º¡¢ÕýÏà¼äʱ£¬¼´

p11?0,p11p21p12?0,p22PΪ¸º¶¨¾ØÕó£¬ÔòV(x)¸º¶¨¡£ÈôÖ÷×ÓÐÐÁÐʽº¬ÓеÈÓÚÁãµÄÇé¿ö£¬ÔòV(x)ΪÕý°ë¶¨»ò

¸º°ë¶¨¡£²»ÊôÒÔÉÏËùÓÐÇé¿öµÄV(x)²»¶¨¡£

3.3.4 ÀîÑÅÆÕŵ·òµÚ¶þ·¨

ÓÉÁ¦Ñ§¾­µäÀíÂÛ¿ÉÖª£¬¶ÔÓÚÒ»¸öÕñ¶¯ÏµÍ³£¬µ±ÏµÍ³×ÜÄÜÁ¿£¨Õý¶¨º¯Êý£©Á¬Ðø¼õС

£¨ÕâÒâζ×Å×ÜÄÜÁ¿¶Ôʱ¼äµÄµ¼Êý±ØÈ»ÊǸº¶¨µÄ£©£¬Ö±µ½Æ½ºâ״̬ʱΪֹ£¬ÔòÕñÔòϵͳÊÇÎȶ¨µÄ¡£

LyapunovµÚ¶þ·¨Êǽ¨Á¢ÔÚ¸üΪÆÕ±éµÄÇé¿öÖ®Éϵ쬼´£ºÈç¹ûϵͳÓÐÒ»¸ö½¥½üÎȶ¨µÄƽºâ״̬£¬Ôòµ±ÆäÔ˶¯µ½Æ½ºâ״̬µÄÎüÒýÓòÄÚʱ£¬ÏµÍ³´æ´¢µÄÄÜÁ¿Ëæ×Åʱ¼äµÄÔö³¤¶øË¥¼õ£¬Ö±µ½ÔÚÆ½ÎÈ״̬´ïµ½¼«Ð¡ÖµÎªÖ¹¡£È»¶ø¶ÔÓÚһЩ´¿Êýѧϵͳ£¬±Ï¾¹»¹Ã»ÓÐÒ»¸ö¶¨Òå¡°ÄÜÁ¿º¯Êý¡±µÄ¼ò±ã·½·¨¡£ÎªÁ˿˷þÕâ¸öÀ§ÄÑ£¬LyapunovÒý³öÁËÒ»¸öÐé¹¹µÄÄÜÁ¿º¯Êý£¬³ÆÎªLyapunovº¯Êý¡£µ±È»£¬Õâ¸öº¯ÊýÎÞÒɱÈÄÜÁ¿¸üΪһ°ã£¬²¢ÇÒÆäÓ¦ÓÃÒ²¸ü¹ã·º¡£Êµ¼ÊÉÏ£¬ÈÎÒ»´¿Á¿º¯ÊýÖ»ÒªÂú×ãLyapunovÎȶ¨ÐÔ¶¨ÀíµÄ¼ÙÉèÌõ¼þ£¬¶¼¿É×÷ΪLyapunovº¯Êý¡£

Lyapunovº¯ÊýÓëx1,x2,?,xnºÍtÓйأ¬ÎÒÃÇÓÃV(x1,x2,?,xn,t)»òÕßV(x,t)À´±íʾLyapunovº¯Êý¡£Èç¹ûÔÚLyapunovº¯ÊýÖв»º¬t,ÔòÓÃV(x1,x2,?,xn)»òV(x)±íʾ¡£ÔÚ

?(x,t)?dV(x,t)/dtµÄ·ûºÅÌØÕ÷£¬ÌṩÁËLyapunovµÚ¶þ·¨ÖУ¬V(x,t)ºÍÆä¶Ôʱ¼äµÄµ¼ÊýVÅÐ¶ÏÆ½ºâ״̬´¦µÄÎȶ¨ÐÔ¡¢½¥½üÎȶ¨ÐÔ»ò²»Îȶ¨ÐÔµÄ×¼Ôò£¬¶ø²»±ØÖ±½ÓÇó³ö·½³ÌµÄ½â£¨ÕâÖÖ·½·¨¼ÈÊÊÓÃÓÚÏßÐÔϵͳ£¬Ò²ÊÊÓÃÓÚ·ÇÏßÐÔϵͳ£©¡£

1¡¢¹ØÓÚ½¥½üÎȶ¨ÐÔ

¿ÉÒÔÖ¤Ã÷£ºÈç¹ûxΪnάÏòÁ¿£¬ÇÒÆä´¿Á¿º¯ÊýV(x)Õý¶¨£¬ÔòÂú×ã

V(x)?C

µÄ״̬x´¦ÓÚnά״̬¿Õ¼äµÄ·â±Õ³¬ÇúÃæÉÏ£¬ÇÒÖÁÉÙ´¦ÓÚÔ­µã¸½½ü£¬Ê½ÖÐCÊÇÕý³£Êý¡£Ëæ×Åx??£¬ÉÏÊö·â±ÕÇúÃæ¿ÉÀ©Õ¹ÎªÕû¸ö״̬¿Õ¼ä¡£Èç¹ûC1?C2£¬Ôò³¬ÇúÃæV(x)?C1ÍêÈ«´¦ÓÚ³¬ÇúÃæV(x)?C2µÄÄÚ²¿¡£

¶ÔÓÚ¸ø¶¨µÄϵͳ£¬Èô¿ÉÇóµÃÕý¶¨µÄ´¿Á¿º¯ÊýV(x)£¬²¢Ê¹ÆäÑØ¹ì¼£¶Ôʱ¼äµÄµ¼Êý×Ü

Ϊ¸ºÖµ£¬ÔòËæ×Åʱ¼äµÄÔö¼Ó£¬V(x)½«È¡Ô½À´Ô½Ð¡µÄCÖµ¡£Ëæ×Åʱ¼äµÄ½øÒ»²½Ôö³¤£¬×îÖÕ

V(x)±äΪÁ㣬¶øxÒ²Ç÷ÓÚÁã¡£ÕâÒâζ×Å£¬×´Ì¬¿Õ¼äµÄÔ­µãÊǽ¥½üÎȶ¨µÄ¡£LyapunovÖ÷ÎÈ

¶¨ÐÔ¶¨Àí¾ÍÊÇǰÊöÊÂʵµÄÆÕ±é»¯£¬Ëü¸ø³öÁ˽¥½üÎȶ¨µÄ³äÒªÌõ¼þ¡£¸Ã¶¨Àí²ûÊöÈçÏ£º ¶¨Àí3.1 (Lyapunov, Ƥ¶ûÏ£µÂ˹»ù£¬°Í°ÍÐÁ£¬¿ËÀ­Ë÷·ò˹»ù) ¿¼ÂÇÈçÏ·ÇÏßÐÔϵͳ

?(t)?f(x(t),t) xʽÖÐ

f(0,t)?0, ¶ÔËùÓÐt?t0

Èç¹û´æÔÚÒ»¸ö¾ßÓÐÁ¬ÐøÒ»½×Æ«µ¼ÊýµÄ´¿Á¿º¯ÊýV(x,t)£¬ÇÒÂú×ãÒÔÏÂÌõ¼þ£º 1¡¢V(x,t)Õý¶¨£» 2¡¢V(x,t)¸º¶¨

ÔòÔÚÔ­µã´¦µÄƽºâ״̬ÊÇ£¨Ò»Ö£©½¥½üÎȶ¨µÄ¡£

½øÒ»²½£¬Èôx??£¬V(x,t)??£¬ÔòÔÚÔ­µã´¦µÄƽºâ״̬ÊÇ´ó·¶Î§Ò»Ö½¥½üÎÈ

¶¨µÄ¡£

------------------------------------------------------------------ Àý3.3 ¿¼ÂÇÈçÏ·ÇÏßÐÔϵͳ

2?1?x2?x1(x12?x2x)?2??x1?x2(x?x)xÈç¹û¶¨ÒåÒ»¸öÕý¶¨´¿Á¿º¯ÊýV(x)

2122

ÏÔȻԭµã£¨x1?0£¬x2?0£©ÊÇΨһµÄƽºâ״̬¡£ÊÔÈ·¶¨ÆäÎȶ¨ÐÔ¡£

22?1?2x2x?2?2(x12?x2V(x)?2x1x)

ÊǸº¶¨µÄ£¬Õâ˵Ã÷V(x)ÑØÈÎÒ»¹ì¼£Á¬ÐøµØ¼õС£¬Òò´ËV(x)ÊÇÒ»¸öLyapunovº¯Êý¡£ÓÉÓÚ

V(x)ËæxÆ«ÀëÆ½ºâ״̬Ç÷ÓÚÎÞÇî¶ø±äΪÎÞÇÔò°´ÕÕ¶¨Àí5.1£¬¸ÃϵͳÔÚÔ­µã´¦µÄƽºâ×´

̬ÊÇ´ó·¶Î§½¥½üÎȶ¨µÄ¡£

×¢Ò⣬ÈôʹV(x)ȡһϵÁеij£Öµ0,C1,C2,?£¨0?C1?C2??£©£¬ÔòV(x)=0

¶ÔÓ¦ÓÚ×´Ì¬Æ½ÃæµÄÔ­µã£¬¶øV(x)?C1£¬V(x)?C2£¬¡­£¬ÃèÊöÁ˰üÎ§×´Ì¬Æ½ÃæÔ­µãµÄ»¥²»ÏཻµÄÒ»´ØÔ²£¬Èçͼ3.2Ëùʾ¡£»¹Ó¦×¢Ò⣬ÓÉÓÚV(x)ÔÚ¾¶ÏòÊÇÎÞ½çµÄ£¬¼´Ëæ×Å

x??£¬V(x)??£¬ËùÒÔÕâÒ»´ØÔ²¿ÉÀ©Õ¹µ½Õû¸ö×´Ì¬Æ½Ãæ¡£

ÓÉÓÚÔ²V(x)?CkÍêÈ«´¦ÔÚV(x)?Ck?1µÄÄÚ²¿£¬ËùÒÔµäÐ͹켣´ÓÍâÏòÀïͨ¹ýVÔ²µÄ±ß½ç¡£Òò´ËLyapunovº¯ÊýµÄ¼¸ºÎÒâÒå¿É²ûÊöÈçÏÂV(x)±íʾ״̬xµ½×´Ì¬¿Õ¼äÔ­µã¾àÀëµÄÒ»ÖÖ¶È

?(x(t))?0£©£¬ÔòÁ¿¡£Èç¹ûÔ­µãÓë˲ʱ״̬x(t)Ö®¼äµÄ¾àÀëËætµÄÔö¼Ó¶øÁ¬ÐøµØ¼õС£¨¼´Vx(t)?0¡£

ͼ3.2 ³£ÊýVÔ²ºÍµäÐ͹켣

------------------------------------------------------------------

¶¨Àí3.1ÊÇLyapunovµÚ¶þ·¨µÄ»ù±¾¶¨Àí£¬ÏÂÃæ¶ÔÕâÒ»ÖØÒª¶¨Àí×÷¼¸µã˵Ã÷¡£ (1) ÕâÀï½ö¸ø³öÁ˳ä·ÖÌõ¼þ£¬Ò²¾ÍÊÇ˵£¬Èç¹ûÎÒÃǹ¹Ôì³öÁËLyapunovº¯ÊýV(x,t)£¬ÄÇôϵͳÊǽ¥½üÎȶ¨µÄ¡£µ«Èç¹ûÎÒÃÇÕÒ²»µ½ÕâÑùµÄLyapunovº¯Êý£¬ÎÒÃDz¢²»Äܸø³öÈκνáÂÛ£¬ÀýÈçÎÒÃDz»ÄܾݴË˵¸ÃϵͳÊDz»Îȶ¨µÄ¡£