150:
74:
53:
84:
22:
725:
Nonsense. Counter-example given in this section is correct, but the conclusions are not. My point is that V must have global minima in origin. Functions of type W = V + const will also prove stability and they are also sometimes called 'Lyapunov functions'. It is vital to give either some references
243:
Hi, in the definition don't we need to mention the system dynamics being considered are ordinary differential equations of the form xdot=f(x) (i.e. not time varying or other dynamics), and that they must be "well behaved" (e.g. f satisfies a
Lipschitz condition within a ball of radius epsilon of the
1400:
The introduction talks about this but there is no description, illustration or reference in the further part of the article. As far as I know there is no such thing: the 3 body problem occurs in a conservative system whereas systems showing
Lyapunov stability are non-conservative systems. I deleted
980:
It's commonly sub-chapter in the
Lyapunov stability section of control textbooks like Slotine's. I rephrased a bit to make the connection more readily obvious to Lyapunov stability and explicitly cite Slotine who's example was already in the section. Given its probably too small to be its own page,
225:
As you can see (gotta find the reference to put it in the article too) in LaSalle's book on
Stability, if you do make a change of variables based on a "translation by y(t)", then y(t) is the origin and a equilibrium point for the new system, and the definition just given in the article is applied.
187:
I did not write the page but know there have been attempts to apply it to forced systems. For example, I.G. Malkin in his book 'Stability of Motion' (Russian but it has been translated to
English) proves that a asymptotically stable system remains stable under disturbances which are bounded if the
1360:
Point 1 of the definition sort of contradicts the lede, which discusses that this applies for a SMALL epsilon , rather than for ANY epsilon. This becomes a problem in the case of a
Lyapunov Orbit which is a special example of a Lissajous Orbit. Using these terms, both have well defined and well
221:
The concept of
Lyapunov Stability is, actually, original from the study of trajectories from ODEs. Then, you should define when a solution of a ODE is Lyapunov Stable. It's a rather similar definition to the one on the page, but since the solution need not to be a equilibrium point anymore, you
1401:
the section but was promptly reverted without explanation or reference to literature which seems to me rather impolite and against
Knowledge policy. The heading to this page says 'Encycopedic content must be verifiable through citations to reliable sources' Is it possible to insist on this?
229:
Although we do not need to specify the kind of system for the definitions, it's very important to realize that a great part of classic results and theorems applies only to unforced systems (actually, I'd use "conservative" systems, since that's the context I do work with them everyday...)
1042:
In the simplest case the origin is much rather 'in' equilibrium, and/or 'is' an equilibrium point. Also, in the simplest case, this is where the first derivative of position i.e. velocity is equal to 0. So, simply put, it is not moving in time and that is all that we mean here
1423:
Sorry, your edit looked like vandalism since you left your signature in place of a paragraph. I am not an expert on this so I can't comment on the content. Feel free to search for a reference or delete the unreferenced paragraph again (without signatures!). Thank you
1001:
I never heard the term 'iterated' to refer to 'discrete-time' or just 'discrete' dynamics in the control literature. Though correct, I would suggest that anyone having a sounder background than me should consider replacing 'iterated' with 'discrete-time'. --
521:
I agree. To use a
Lyapunov function itself to prove asymptotic stability (without resorting to the invariance principle or Barbalat's Lemma), you have to choose a different function. I suggest changing the section to use a simpler first order system.
248:"entirely in the plane of the two primary bodies" Two points don't define a plane. I think what is meant is the plane of the orbits of the two bodies, which in most cases means the plane of the orbit of the smaller body around the larger.
214:
First, I've already altered the definition here (I didn't have a login at that time, sorry... when I find out how, I promise to put my IP here), and basically that's the one present now, when it comes to &epsillon; and δ.
1017:(Good article.) Forgive a naive question, but what kind of "equilibrium" is assumed, supposedly WLOG, at the origin? The term is not defined. Do you mean zero sum of forces? Stable orbits? Some other dynamical invariant?
1162:
1203:
Exponential stability doesn't require asymptotic stability. If the equilibrium is exponentially stable it is also stable (we can put δ=ε/α) and obviously the solutions from the δ-neighbourhood converge to it.
255:
718:, and that the term 'candidate' is used to distinguish functions which have yet to be proven to be Lyapunov functions (candidate Lyapunov functions) from those which have been proven.
286:
It's true that the closed-loop system is unforced mathematically, but from the perspective of the original system, we have a system that was stabilized using the control force.
453:
218:
But the article is, actually, wrong in the following sense: we're dealing with
Lyapunov Stability of Equilibrium Points. There's no need to specify if the system is unforced.
931:
645:
516:
382:
344:
486:
680:
580:
866:
765:
409:
712:
1490:
1302:
1298:
1284:
839:
158:
818:
1164:
has an attractive equilibrium at (x,y)=(1,0) which is not Lyapunov stable. I think this case is impossible for hyperbolic equilibria, but I'm not sure. --
518:. To prove asymptotic stability, either a new Lyapunov candidate function needs to be considered, or LaSalle's invariance principal needs to be applied.
1073:
Asymptotic stability implies Lyapunov stability only since it is explicitly included in the definition. E.g. the system given in polar coordinates as
1051:
It is redundant to define asymptotic stability as being Lyapunov stable, since it can be shown that asymptotic stability implies Lyapunov stability.
1485:
140:
130:
308:
In the Van der Pol oscillator example, we clearly have the origin as an equilibrium point to the system. The problem stems from the "claim" that
1250:
1495:
1480:
1381:
1184:
I can't find it here, nor in the dynamic system page. I'm assuming it's just another name for fixpoint, but some definition would be handy.
1211:
1185:
1058:
1165:
1076:
727:
259:
1032:
97:
58:
1457:
1345:
33:
222:
should compare the norm of the difference between y(t) (the solution you're studing) and x(t) as in the article.
1301:
to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the
1385:
1389:
1251:
https://web.archive.org/web/20071018132250/http://mne.ksu.edu:80/research/laboratories/non-linear-controls-lab
1215:
1189:
1062:
868:, is a condition for stability. So there must be global minimum in the equilibrium (which is not the case for
21:
1169:
233:
Well, sorry for anything! This is my first writing here... so I'll try to learn it better how to contribute!
1453:
1336:
1242:
731:
1036:
89:
1320:
If you have discovered URLs which were erroneously considered dead by the bot, you can report them with
1308:
183:
You who wrote this page: are you sure that we cannot apply the Lyapunov stability to forced systems????
39:
1254:
1241:. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit
986:
211:
I'm not in the mood to put it all correctly right now, but I'd like at least to comment some details.
1377:
1275:
1207:
1054:
1020:
938:
934:
251:
1369:
Shouldn't the first mention of Aleksandr Lyapunov link to the wikipedia page about him? Like this:
414:
149:
1429:
971:
871:
585:
963:
523:
491:
349:
311:
237:
1370:
1234:
1305:
before doing mass systematic removals. This message is updated dynamically through the template
740:
The course notes I have on this subject (not publicly accessible) state that, given equilibrium
1321:
280:
is unstable when f(t) = 0, but stable when f(t) = -5*x. The closed-loop system in this case is
1028:
1006:
458:
1328:
650:
550:
1443:
1406:
844:
743:
387:
193:
106:
688:
274:
Can't Lyapunov stability apply to forced systems, like closed-loop systems for example?
73:
52:
1425:
1287:, "External links modified" talk page sections are no longer generated or monitored by
967:
770:
295:
1327:
If you found an error with any archives or the URLs themselves, you can fix them with
824:
1474:
953:
719:
956:
1024:
1294:
1003:
982:
1439:
1418:
1402:
1293:. No special action is required regarding these talk page notices, other than
189:
79:
1438:
Yes it was a bit messy. But why edit on a subject you are not familiar with?
1461:
1447:
1433:
1410:
1350:
1219:
1193:
1173:
1066:
990:
975:
942:
735:
526:
298:
263:
197:
952:
This looks like a really cute lemma, but does it belong on this page?
102:
1255:
http://www.mne.ksu.edu/research/laboratories/non-linear-controls-lab
981:
unless other pages refer back to it, I think it is a good section.
188:
bound is sufficiently small. But there is no estimate of how small.
1157:{\displaystyle {\dot {r}}=r(1-r),{\dot {\theta }}=\sin(\theta /2)}
15:
411:
contribution, this equation is negative semi-definite, i.e.
1260:
When you have finished reviewing my changes, please set the
244:
origin)? Also you might want to mention the norms involved.
148:
1245:
for additional information. I made the following changes:
536:
well a short bibliography would be useful, wouldn't it?
1238:
1079:
962:
By the way, it is named after Romanian mathematician
874:
847:
827:
774:
746:
691:
653:
588:
553:
494:
461:
417:
390:
352:
314:
1297:using the archive tool instructions below. Editors
1156:
925:
860:
833:
811:
759:
706:
674:
639:
574:
510:
480:
447:
403:
376:
338:
304:The Van der Pol oscillator example is Incorrect.
1364:
1047:Asymptotic Stability Implies Lyapunov Stability
1283:This message was posted before February 2018.
8:
101:, which collaborates on articles related to
1012:
19:
1375:
1233:I have just modified one external link on
1205:
249:
47:
1143:
1117:
1116:
1081:
1080:
1078:
915:
907:
893:
873:
852:
846:
826:
800:
773:
751:
745:
690:
652:
629:
621:
607:
587:
552:
502:
493:
466:
460:
419:
418:
416:
395:
389:
354:
353:
351:
316:
315:
313:
1396:Lyapunov stability in the 3 body problem
685:Also, my understanding was that such a
49:
294:In the 9th line the 0 is not the EP.--
256:2601:644:8D7F:FD10:9C6C:1FEE:454F:3FE9
1491:Systems articles in dynamical systems
1272:to let others know (documentation at
7:
543:Lyapunov second theorem on stability
157:This article is within the field of
95:This article is within the scope of
1199:Definition of exponential stability
38:It is of interest to the following
1365:'Aleksandr Lyapunov' is not a link
495:
270:forced systems can be stable ITSOL
14:
1237:. Please take a moment to review
82:
72:
51:
20:
1486:Mid-importance Systems articles
996:
582:? The definition given allows
448:{\displaystyle {\dot {V}}(x)=0}
135:This article has been rated as
1151:
1137:
1110:
1098:
1013:'the origin is an equilibrium'
926:{\displaystyle V(x)=1/(1+|x|)}
920:
916:
908:
898:
884:
878:
806:
793:
784:
778:
701:
695:
663:
657:
640:{\displaystyle V(x)=1/(1+|x|)}
634:
630:
622:
612:
598:
592:
563:
557:
436:
430:
371:
365:
333:
327:
1:
1067:15:49, 18 November 2009 (UTC)
957:22:21, 24 February 2007 (UTC)
736:17:58, 25 November 2010 (UTC)
547:Isn't it also required that
511:{\displaystyle \forall x_{1}}
377:{\displaystyle {\dot {V}}(x)}
339:{\displaystyle {\dot {V}}(x)}
299:18:57, 11 February 2006 (UTC)
264:19:18, 19 December 2021 (UTC)
198:17:03, 21 December 2010 (UTC)
115:Knowledge:WikiProject Systems
1496:WikiProject Systems articles
1481:Start-Class Systems articles
1356:Problem with the definition?
1351:14:02, 9 November 2016 (UTC)
1194:20:39, 1 February 2014 (UTC)
1174:13:28, 8 December 2013 (UTC)
722:19:00, 31 January 2007(UTC)
346:is negative definite. Since
118:Template:WikiProject Systems
1448:16:21, 8 January 2022 (UTC)
1434:16:43, 7 January 2022 (UTC)
1411:14:33, 7 January 2022 (UTC)
1361:bounded values of epsilon.
1037:18:26, 17 August 2008 (UTC)
1007:12:10, 31 August 2007 (UTC)
943:13:24, 14 August 2014 (UTC)
1512:
1462:08:20, 19 April 2023 (UTC)
1314:(last update: 5 June 2024)
1230:Hello fellow Wikipedians,
1220:04:14, 25 April 2016 (UTC)
726:or full proofs. (or both)
682:is asymptotically stable.
527:06:06, 26 April 2007 (UTC)
141:project's importance scale
1452:I just deleted it again.
1390:04:30, 3 March 2018 (UTC)
991:19:45, 14 June 2024 (UTC)
976:08:40, 25 June 2015 (UTC)
240:22:40, May 5, 2005 (UTC)
156:
134:
67:
46:
812:{\displaystyle V(x): -->
1226:External links modified
1180:What is an equilibrium?
481:{\displaystyle x_{2}=0}
1158:
968:Heinrich ⅩⅦ von Bayern
927:
862:
835:
814:
761:
708:
676:
675:{\displaystyle x(t)=t}
641:
576:
575:{\displaystyle V(0)=0}
512:
482:
449:
405:
378:
340:
153:
90:Systems science portal
28:This article is rated
1159:
947:
928:
863:
861:{\displaystyle x_{e}}
836:
815:
762:
760:{\displaystyle x_{e}}
709:
677:
642:
577:
513:
483:
450:
406:
404:{\displaystyle x_{1}}
379:
341:
152:
1295:regular verification
1077:
872:
845:
825:
772:
744:
707:{\displaystyle V(x)}
689:
651:
586:
551:
492:
459:
415:
388:
350:
312:
1285:After February 2018
1264:parameter below to
277:dx/dt - 4*x = f(t)
98:WikiProject Systems
1454:The-erinaceous-one
1371:Aleksandr Lyapunov
1339:InternetArchiveBot
1290:InternetArchiveBot
1235:Lyapunov stability
1154:
997:'Iterated' systems
923:
858:
831:
809:
757:
704:
672:
637:
572:
532:lack of references
508:
478:
445:
401:
374:
336:
154:
34:content assessment
1392:
1380:comment added by
1315:
1222:
1210:comment added by
1125:
1089:
1057:comment added by
1039:
1023:comment added by
834:{\displaystyle x}
819:V(x_{e})}" /: -->
716:Lyapunov function
427:
362:
324:
290:Equilibrium point
266:
254:comment added by
173:
172:
169:
168:
165:
164:
159:Dynamical systems
1503:
1422:
1349:
1340:
1313:
1312:
1291:
1279:
1163:
1161:
1160:
1155:
1147:
1127:
1126:
1118:
1091:
1090:
1082:
1069:
1018:
948:Barbalat's lemma
932:
930:
929:
924:
919:
911:
897:
867:
865:
864:
859:
857:
856:
840:
838:
837:
832:
820:
817:
816:
810:
805:
804:
766:
764:
763:
758:
756:
755:
713:
711:
710:
705:
681:
679:
678:
673:
646:
644:
643:
638:
633:
625:
611:
581:
579:
578:
573:
517:
515:
514:
509:
507:
506:
487:
485:
484:
479:
471:
470:
454:
452:
451:
446:
429:
428:
420:
410:
408:
407:
402:
400:
399:
383:
381:
380:
375:
364:
363:
355:
345:
343:
342:
337:
326:
325:
317:
123:
122:
121:Systems articles
119:
116:
113:
92:
87:
86:
85:
76:
69:
68:
63:
55:
48:
31:
25:
24:
16:
1511:
1510:
1506:
1505:
1504:
1502:
1501:
1500:
1471:
1470:
1416:
1398:
1393:
1382:121.212.147.109
1367:
1358:
1343:
1338:
1306:
1299:have permission
1289:
1273:
1243:this simple FaQ
1228:
1201:
1182:
1075:
1074:
1052:
1049:
1015:
999:
950:
870:
869:
848:
843:
842:
823:
822:
796:
771:V(x_{e})}": -->
769:
768:
747:
742:
741:
687:
686:
649:
648:
584:
583:
549:
548:
545:
534:
498:
490:
489:
462:
457:
456:
413:
412:
391:
386:
385:
348:
347:
310:
309:
306:
292:
272:
206:
204:Overall coments
181:
120:
117:
114:
111:
110:
107:systems science
88:
83:
81:
61:
32:on Knowledge's
29:
12:
11:
5:
1509:
1507:
1499:
1498:
1493:
1488:
1483:
1473:
1472:
1469:
1468:
1467:
1466:
1465:
1464:
1397:
1394:
1374:
1366:
1363:
1357:
1354:
1333:
1332:
1325:
1258:
1257:
1249:Added archive
1227:
1224:
1212:91.122.175.225
1200:
1197:
1186:62.245.100.121
1181:
1178:
1177:
1176:
1153:
1150:
1146:
1142:
1139:
1136:
1133:
1130:
1124:
1121:
1115:
1112:
1109:
1106:
1103:
1100:
1097:
1094:
1088:
1085:
1059:130.15.101.152
1048:
1045:
1014:
1011:
998:
995:
994:
993:
978:
949:
946:
922:
918:
914:
910:
906:
903:
900:
896:
892:
889:
886:
883:
880:
877:
855:
851:
830:
808:
803:
799:
795:
792:
789:
786:
783:
780:
777:
754:
750:
703:
700:
697:
694:
671:
668:
665:
662:
659:
656:
647:to prove that
636:
632:
628:
624:
620:
617:
614:
610:
606:
603:
600:
597:
594:
591:
571:
568:
565:
562:
559:
556:
544:
541:
539:
533:
530:
505:
501:
497:
477:
474:
469:
465:
444:
441:
438:
435:
432:
426:
423:
398:
394:
373:
370:
367:
361:
358:
335:
332:
329:
323:
320:
305:
302:
291:
288:
283:dx/dt + x = 0
271:
268:
246:
205:
202:
201:
200:
180:
177:
175:
171:
170:
167:
166:
163:
162:
155:
145:
144:
137:Mid-importance
133:
127:
126:
124:
94:
93:
77:
65:
64:
62:Mid‑importance
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
1508:
1497:
1494:
1492:
1489:
1487:
1484:
1482:
1479:
1478:
1476:
1463:
1459:
1455:
1451:
1450:
1449:
1445:
1441:
1437:
1436:
1435:
1431:
1427:
1420:
1415:
1414:
1413:
1412:
1408:
1404:
1395:
1391:
1387:
1383:
1379:
1373:
1372:
1362:
1355:
1353:
1352:
1347:
1342:
1341:
1330:
1326:
1323:
1319:
1318:
1317:
1310:
1304:
1300:
1296:
1292:
1286:
1281:
1277:
1271:
1267:
1263:
1256:
1252:
1248:
1247:
1246:
1244:
1240:
1236:
1231:
1225:
1223:
1221:
1217:
1213:
1209:
1198:
1196:
1195:
1191:
1187:
1179:
1175:
1171:
1167:
1166:132.68.52.136
1148:
1144:
1140:
1134:
1131:
1128:
1122:
1119:
1113:
1107:
1104:
1101:
1095:
1092:
1086:
1083:
1072:
1071:
1070:
1068:
1064:
1060:
1056:
1046:
1044:
1040:
1038:
1034:
1030:
1026:
1022:
1010:
1008:
1005:
992:
988:
984:
979:
977:
973:
969:
965:
961:
960:
959:
958:
955:
945:
944:
940:
936:
912:
904:
901:
894:
890:
887:
881:
875:
853:
849:
828:
801:
797:
790:
787:
781:
775:
752:
748:
738:
737:
733:
729:
723:
721:
717:
698:
692:
683:
669:
666:
660:
654:
626:
618:
615:
608:
604:
601:
595:
589:
569:
566:
560:
554:
542:
540:
537:
531:
529:
528:
525:
519:
503:
499:
475:
472:
467:
463:
442:
439:
433:
424:
421:
396:
392:
368:
359:
356:
330:
321:
318:
303:
301:
300:
297:
289:
287:
284:
281:
278:
275:
269:
267:
265:
261:
257:
253:
245:
241:
239:
234:
231:
227:
223:
219:
216:
212:
209:
203:
199:
195:
191:
186:
185:
184:
178:
176:
160:
151:
147:
146:
142:
138:
132:
129:
128:
125:
108:
104:
100:
99:
91:
80:
78:
75:
71:
70:
66:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
1399:
1376:— Preceding
1368:
1359:
1337:
1334:
1309:source check
1288:
1282:
1269:
1265:
1261:
1259:
1232:
1229:
1206:— Preceding
1202:
1183:
1050:
1041:
1016:
1000:
964:Ion Barbalat
951:
739:
728:95.67.191.95
724:
715:
714:is called a
684:
546:
538:
535:
520:
307:
293:
285:
282:
279:
276:
273:
250:— Preceding
247:
242:
235:
232:
228:
224:
220:
217:
213:
210:
207:
182:
174:
136:
96:
40:WikiProjects
1276:Sourcecheck
1053:—Preceding
1019:—Preceding
179:Application
30:Start-class
1475:Categories
1346:Report bug
935:Zeebrommer
1426:Ita140188
1329:this tool
1322:this tool
813:V(x_{e})}
296:Pokipsy76
1378:unsigned
1335:Cheers.—
1208:unsigned
1055:unsigned
1033:contribs
1021:unsigned
954:LachlanA
821:for all
720:LachlanA
252:unsigned
1262:checked
1239:my edit
1025:Dratman
841:except
524:S280Z28
384:has no
238:Rfreire
139:on the
112:Systems
103:systems
59:Systems
1270:failed
1004:Biscay
983:Treyra
36:scale.
1440:JFB80
1419:JFB80
1403:JFB80
788:: -->
455:when
190:JFB80
1458:talk
1444:talk
1430:talk
1407:talk
1386:talk
1266:true
1216:talk
1190:talk
1170:talk
1063:talk
1029:talk
987:talk
972:talk
939:talk
732:talk
488:and
260:talk
208:Hi,
194:talk
105:and
1303:RfC
1280:).
1268:or
1253:to
1132:sin
966:. -
131:Mid
1477::
1460:)
1446:)
1432:)
1424:--
1409:)
1388:)
1316:.
1311:}}
1307:{{
1278:}}
1274:{{
1218:)
1192:)
1172:)
1141:θ
1135:
1123:˙
1120:θ
1105:−
1087:˙
1065:)
1035:)
1031:•
1009:.
989:)
974:)
941:)
933:)
767:,
734:)
496:∀
425:˙
360:˙
322:˙
262:)
236:--
196:)
1456:(
1442:(
1428:(
1421::
1417:@
1405:(
1384:(
1348:)
1344:(
1331:.
1324:.
1214:(
1188:(
1168:(
1152:)
1149:2
1145:/
1138:(
1129:=
1114:,
1111:)
1108:r
1102:1
1099:(
1096:r
1093:=
1084:r
1061:(
1027:(
985:(
970:(
937:(
921:)
917:|
913:x
909:|
905:+
902:1
899:(
895:/
891:1
888:=
885:)
882:x
879:(
876:V
854:e
850:x
829:x
807:)
802:e
798:x
794:(
791:V
785:)
782:x
779:(
776:V
753:e
749:x
730:(
702:)
699:x
696:(
693:V
670:t
667:=
664:)
661:t
658:(
655:x
635:)
631:|
627:x
623:|
619:+
616:1
613:(
609:/
605:1
602:=
599:)
596:x
593:(
590:V
570:0
567:=
564:)
561:0
558:(
555:V
504:1
500:x
476:0
473:=
468:2
464:x
443:0
440:=
437:)
434:x
431:(
422:V
397:1
393:x
372:)
369:x
366:(
357:V
334:)
331:x
328:(
319:V
258:(
192:(
161:.
143:.
109:.
42::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
