740:
identified such indifference points for the first time in 1977. Further, Skiba, Sethi, and
Deckert and Nishimura explored these indifference points in economic models. The term DNSS (Deckert, Nishimura, Sethi, Skiba) points, introduced by Grass et al., recognizes (alphabetically) the contributions of
34:
problems that exhibit multiple optimal solutions. A Sethi-Skiba point is an indifference point in an optimal control problem such that starting from such a point, the problem has more than one different optimal solutions. A good discussion of such points can be found in Grass et al.
167:
285:
1430:; Seidl, Andrea; Wrzaczek, Stefan (2022), Boado-Penas, MarĂa del Carmen; Eisenberg, Julia; Ĺžahin, Ĺžule (eds.), "COVID-19 and Optimal LockdownStrategies: The Effect of New and MoreVirulent Strains",
1177:
1287:
806:
862:
1064:
546:
1015:
907:
53:
1210:
1097:
690:
657:
314:
1395:
Sethi, S.P. (2021). " Optimal
Control Theory: Applications to Management Science and Economics". Fourth Edition, Springer Nature Switzerland AG, ISBN 978-3-319-98236-6
412:
940:
472:
1597:
Zeiler, I., Caulkins, J., Grass, D., Tragler, G. (2009). Keeping
Options Open: An Optimal Control Model with Trajectories that Reach a DNSS Point in Positive Time.
717:
624:
593:
1611:
Caulkins, J. P.; Feichtinger, G.; Grass, D.; Tragler, G. (2009). "Optimal control of terrorism and global reputation: A case study with novel threshold behavior".
969:
372:
343:
566:
492:
452:
432:
392:
1856:
Deckert, D.W.; Nishimura, K. (1983). "A Complete
Characterization of Optimal Growth Paths in an Aggregated Model with Nonconcave Production Function".
1565:
1459:
1410:
178:
1893:
1581:
1475:
1427:
1314:
1102:
599:
if the system starting from it exhibits multiple optimal solutions or equilibria. Thus, at least in the neighborhood of
434:
are assumed to be continuously differentiable with respect to their arguments and they do not depend explicitly on time
1215:
729:
Some references on the applications of Sethi-Skiba points are
Caulkins et al., Zeiler et al., and Carboni and Russu
1653:"Taxation, Corruption and Punishment: Integrating Evolutionary Game into the Optimal Control of Government Policy"
760:
1888:
810:
1020:
497:
974:
1693:(1977). "Nearest Feasible Paths in Optimal Control Problems: Theory, Examples, and Counterexamples".
1637:
I. Zeiler, J. P. Caulkins, and G. Tragler. When Two Become One: Optimal
Control of Interacting Drug.
1491:
162:{\displaystyle \max _{u(t)\in \Omega }\int _{0}^{\infty }e^{-\rho t}\varphi \left(x(t),u(t)\right)dt}
1652:
867:
1819:
1745:
1710:
1182:
1069:
662:
629:
293:
1844:
1784:
1672:
1561:
1527:
1509:
1455:
1426:
Caulkins, Jonathan P.; Grass, Dieter; Feichtinger, Gustav; Hartl, Richard F.; Kort, Peter M.;
1406:
1366:
1348:
1313:
Caulkins, Jonathan P.; Grass, Dieter; Feichtinger, Gustav; Hartl, Richard F.; Kort, Peter M.;
397:
912:
457:
1865:
1811:
1776:
1737:
1702:
1664:
1620:
1517:
1499:
1445:
1435:
1398:
1356:
1338:
1330:
1765:"Nearest feasible paths in optimal control problems: Theory, examples, and counterexamples"
719:
is an indifference point from which the system could move to either of the two equilibria.
695:
602:
571:
1690:
1558:
Optimal
Control of Nonlinear Processes: With Applications in Drugs, Corruption, and Terror
945:
737:
723:
348:
319:
44:
31:
1495:
1474:
Caulkins, Jonathan; Grass, Dieter; Feichtinger, Gustav; Hartl, Richard; Kort, Peter M.;
1522:
1479:
1361:
1318:
551:
477:
437:
417:
377:
1882:
1869:
1823:
1714:
1343:
494:. Furthermore, it is assumed that the integral converges for any admissible solution
726:
problems, Grass et al. and Zeiler et al. present examples that exhibit DNSS curves.
1334:
1836:
Sethi, S.P., "Optimal
Quarantine Programmes for Controlling an Epidemic Spread,"
1556:
Grass, D.; Caulkins, J. P.; Feichtinger, G.; Tragler, G.; Behrens, D. A. (2008).
1504:
1434:, Springer Actuarial, Cham: Springer International Publishing, pp. 163–190,
1728:
Skiba, A.K. (1978). "Optimal Growth with a Convex-Concave
Production Function".
1440:
1668:
1624:
1402:
474:
is the set of feasible controls and it also is explicitly independent of time
1788:
1676:
1513:
1352:
1584:, Seidl, A., Wrzaczek, A. (2020). “When should the Covid-19 lockdown end?”.
1390:
1802:
Sethi, S.P. (1979). "Optimal
Advertising Policy with the Contagion Model".
1531:
1370:
1297:
For further details and extensions, the reader is referred to Grass et al.
17:
1580:
Caulkins, J. P., Grass, D., Feichtinger, G., Hartl, R. F., Kort, P. M.,
1450:
1841:
1815:
1780:
1749:
1706:
741:
these authors. These indifference points have been also referred to as
1764:
1741:
47:
problems that are autonomous. These problems can be formulated as
942:
is a Sethi-Skiba point for this problem because the optimal path
280:{\displaystyle {\dot {x}}(t)=f\left(x(t),u(t)\right),x(0)=x_{0},}
43:
Of particular interest here are discounted infinite horizon
374:
are the state and control variables, respectively, at time
1601:, Vol. 48, No. 6, pp. 3698-3707.| doi =10.1137/080719741 |
548:. In such a problem with one-dimensional state variable
757:
A simple problem exhibiting this behavior is given by
1218:
1185:
1105:
1072:
1023:
977:
948:
915:
870:
813:
763:
698:
665:
632:
605:
574:
554:
500:
480:
460:
440:
420:
400:
380:
351:
322:
296:
181:
56:
1325:. The economics of epidemics and emerging diseases.
1281:
1204:
1172:{\displaystyle x(t)=-1+e^{-t\left(x_{0}+1\right)}}
1171:
1091:
1058:
1009:
963:
934:
901:
856:
800:
711:
684:
651:
618:
587:
560:
540:
486:
466:
446:
426:
406:
386:
366:
337:
308:
279:
161:
1651:Carboni, Oliviero A.; Russu, Paolo (2021-06-01).
1480:"How long should the COVID-19 lockdown continue?"
1282:{\displaystyle x(t)=1+e^{-t\left(x_{0}-1\right)}}
1641:Vienna University of Technology, Vienna, Austria
1478:; Seidl, Andrea; Wrzaczek, Stefan (2020-12-02).
1317:; Seidl, Andrea; Wrzaczek, Stefan (2021-03-01).
58:
1804:Journal of Optimization Theory and Applications
1769:Journal of Optimization Theory and Applications
1695:Journal of Optimization Theory and Applications
1319:"The optimal lockdown intensity for COVID-19"
8:
801:{\displaystyle \varphi \left(x,u\right)=xu,}
1432:Pandemics: Insurance and Social Protection
626:, the system moves to one equilibrium for
1521:
1503:
1449:
1439:
1360:
1342:
1260:
1244:
1217:
1190:
1184:
1150:
1134:
1104:
1077:
1071:
1042:
1022:
993:
976:
947:
920:
914:
869:
812:
762:
703:
697:
676:
664:
643:
631:
610:
604:
579:
573:
553:
499:
479:
459:
439:
419:
399:
379:
350:
321:
295:
268:
183:
182:
180:
101:
91:
86:
61:
55:
1599:SIAM Journal on Control and Optimization
1838:Journal of Operational Research Society
1551:
1549:
1547:
1545:
1543:
1541:
1305:
857:{\displaystyle f\left(x,u\right)=-x+u,}
1059:{\displaystyle \left(-1+e^{-t}\right)}
541:{\displaystyle \left(x(.),u(.)\right)}
30:, also known as DNSS points, arise in
1010:{\displaystyle \left(1-e^{-t}\right)}
7:
1384:
1382:
1380:
909:. It is shown in Grass et al. that
871:
461:
92:
77:
25:
1323:Journal of Mathematical Economics
1657:International Game Theory Review
1228:
1222:
1115:
1109:
958:
952:
530:
524:
515:
509:
361:
355:
332:
326:
258:
252:
238:
232:
223:
217:
200:
194:
145:
139:
130:
124:
71:
65:
1:
1335:10.1016/j.jmateco.2021.102489
902:{\displaystyle \Omega =\left}
1870:10.1016/0022-0531(83)90081-9
1505:10.1371/journal.pone.0243413
1763:Sethi, S. P. (1977-12-01).
1613:Operations Research Letters
1441:10.1007/978-3-030-78334-1_9
1910:
1858:Journal of Economic Theory
1205:{\displaystyle x_{0}>0}
1092:{\displaystyle x_{0}<0}
685:{\displaystyle x<x_{0}}
652:{\displaystyle x>x_{0}}
309:{\displaystyle \rho >0}
1669:10.1142/S021919892050019X
1625:10.1016/j.orl.2009.07.003
1403:10.1007/978-3-319-98237-3
1389:Sethi, Suresh P. (2021).
1344:10067/1777560151162165141
1840:, 29(3), 1978, 265-268.
1391:"Optimal Control Theory"
407:{\displaystyle \varphi }
935:{\displaystyle x_{0}=0}
749:in earlier literature.
467:{\displaystyle \Omega }
1894:Mathematical economics
1283:
1212:, the optimal path is
1206:
1173:
1099:, the optimal path is
1093:
1060:
1011:
965:
936:
903:
858:
802:
713:
686:
653:
620:
589:
562:
542:
488:
468:
448:
428:
408:
388:
368:
339:
316:is the discount rate,
310:
281:
163:
1284:
1207:
1174:
1094:
1061:
1012:
966:
937:
904:
859:
803:
714:
712:{\displaystyle x_{0}}
687:
654:
621:
619:{\displaystyle x_{0}}
590:
588:{\displaystyle x_{0}}
563:
543:
489:
469:
449:
429:
409:
389:
369:
340:
311:
282:
164:
1216:
1183:
1103:
1070:
1021:
975:
964:{\displaystyle x(t)}
946:
913:
868:
811:
761:
722:For two-dimensional
696:
663:
630:
603:
572:
568:, the initial state
552:
498:
478:
458:
438:
418:
398:
378:
367:{\displaystyle u(t)}
349:
338:{\displaystyle x(t)}
320:
294:
179:
54:
1496:2020PLoSO..1543413C
659:and to another for
96:
1816:10.1007/BF00934454
1781:10.1007/BF00933297
1707:10.1007/BF00933297
1279:
1202:
1169:
1089:
1056:
1007:
961:
932:
899:
854:
798:
709:
682:
649:
616:
585:
558:
538:
484:
464:
444:
424:
404:
384:
364:
335:
306:
277:
159:
82:
81:
28:Sethi-Skiba points
1588:Ausgabe 69: 10-13
1567:978-3-540-77646-8
1476:Prskawetz, Alexia
1461:978-3-030-78334-1
1428:Prskawetz, Alexia
1412:978-3-319-98236-6
1315:Prskawetz, Alexia
692:. In this sense,
597:Sethi-Skiba point
561:{\displaystyle x}
487:{\displaystyle t}
447:{\displaystyle t}
427:{\displaystyle f}
387:{\displaystyle t}
191:
57:
16:(Redirected from
1901:
1874:
1873:
1853:
1847:
1834:
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1827:
1799:
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1608:
1602:
1595:
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1578:
1572:
1571:
1553:
1536:
1535:
1525:
1507:
1490:(12): e0243413.
1471:
1465:
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1423:
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1416:
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1167:
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1082:
1081:
1066:. Note that for
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1016:
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863:
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344:
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245:
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193:
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168:
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165:
160:
152:
148:
112:
111:
95:
90:
80:
21:
1909:
1908:
1904:
1903:
1902:
1900:
1899:
1898:
1889:Optimal control
1879:
1878:
1877:
1855:
1854:
1850:
1835:
1831:
1801:
1800:
1796:
1762:
1761:
1757:
1742:10.2307/1914229
1727:
1726:
1722:
1689:
1688:
1684:
1650:
1649:
1645:
1639:Working paper,
1636:
1632:
1610:
1609:
1605:
1596:
1592:
1579:
1575:
1568:
1555:
1554:
1539:
1473:
1472:
1468:
1462:
1425:
1424:
1420:
1413:
1388:
1387:
1378:
1312:
1311:
1307:
1303:
1295:
1256:
1255:
1251:
1240:
1214:
1213:
1186:
1181:
1180:
1146:
1145:
1141:
1130:
1101:
1100:
1073:
1068:
1067:
1038:
1028:
1024:
1019:
1018:
989:
982:
978:
973:
972:
944:
943:
916:
911:
910:
881:
877:
866:
865:
821:
817:
809:
808:
771:
767:
759:
758:
755:
738:Suresh P. Sethi
735:
724:optimal control
699:
694:
693:
672:
661:
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639:
628:
627:
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569:
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495:
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375:
347:
346:
318:
317:
292:
291:
264:
213:
209:
177:
176:
120:
116:
97:
52:
51:
45:optimal control
41:
32:optimal control
23:
22:
15:
12:
11:
5:
1907:
1905:
1897:
1896:
1891:
1881:
1880:
1876:
1875:
1864:(2): 332–354.
1848:
1829:
1810:(4): 615–627.
1794:
1775:(4): 563–579.
1755:
1736:(3): 527–539.
1720:
1701:(4): 563–579.
1682:
1663:(2): 2050019.
1643:
1630:
1619:(6): 387–391.
1603:
1590:
1573:
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1537:
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1411:
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1111:
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1088:
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1080:
1076:
1054:
1048:
1045:
1041:
1037:
1034:
1031:
1027:
1005:
999:
996:
992:
988:
985:
981:
971:can be either
960:
957:
954:
951:
931:
928:
923:
919:
897:
893:
890:
887:
884:
880:
876:
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847:
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834:
830:
827:
824:
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816:
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791:
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780:
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770:
766:
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751:
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609:
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40:
37:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1906:
1895:
1892:
1890:
1887:
1886:
1884:
1871:
1867:
1863:
1859:
1852:
1849:
1846:
1843:
1842:JSTOR 3009454
1839:
1833:
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1790:
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1696:
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1654:
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1626:
1622:
1618:
1614:
1607:
1604:
1600:
1594:
1591:
1587:
1583:
1582:Prskawetz, A.
1577:
1574:
1569:
1563:
1559:
1552:
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1546:
1544:
1542:
1538:
1533:
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1358:
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1336:
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1324:
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1316:
1309:
1306:
1300:
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1292:
1290:
1273:
1269:
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1261:
1257:
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1248:
1245:
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1237:
1234:
1231:
1225:
1219:
1199:
1196:
1191:
1187:
1163:
1159:
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1147:
1142:
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1135:
1131:
1127:
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1118:
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1845:SSRN 3587573
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1730:Econometrica
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743:Skiba points
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394:, functions
289:
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1691:Sethi, S.P.
1883:Categories
1329:: 102489.
1301:References
1293:Extensions
747:DNS points
39:Definition
18:DNSS point
1824:121398518
1789:1573-2878
1715:123705828
1677:0219-1989
1514:1932-6203
1353:0304-4068
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