1378:(DAE's), which is the number of times the algebraic equations of a DAE must be differentiated in order to obtain a complete system of differential equations for all variables. It is also a notion close to the relative degree of Control Theory, which is, roughly speaking, the number of times an "output" variable has to be differentiated so that an "input" variable appears explicitly in Control Theory this is used to derive a canonical state space form which involves the so-called "zero-dynamics", a fundamental concept for control). For a DVI, the index is the number of differentiations of
22:
1765:
Differential variational inequalities with index greater than two are generally not meaningful, but certain conditions and interpretations can make them meaningful (see the references Acary, Brogliato and
Goeleven, and Heemels, Schumacher, and Weiland below). One crucial step is to first define a
598:
falling from a height towards a table. Assume that the forces acting on the ball are gravitation and the contact forces of the table preventing penetration. Then the differential equation describing the motion is
451:
563:
1761:
For the ideal diode systems, the computations are considerably more difficult, but provided some generally valid conditions hold, the differential variational inequality can be shown to have index one.
290:
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An ideal diode is a diode that conducts electricity in the forward direction with no resistance if a forward voltage is applied, but allows no current to flow in the reverse direction. Then if the
1337:
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and networks of queues (where the constraints can either be upper limits on queue length or that the queue length cannot become negative). DVIs are related to a number of other concepts including
681:
1173:
1606:
1698:
1362:. If the diode is in a circuit containing a memory element, such as a capacitor or inductor, then the circuit can be represented as a differential variational inequality.
856:
917:
1057:
195:
1479:
993:
1206:
952:
319:
1374:
of a DVI is important and determines many questions of existence and uniqueness of solutions to a DVI. This concept is closely related to the concept of index for
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unknown. While the ball and the table are separated, there is no contact force. There cannot be penetration (for a rigid ball and a rigid table), so
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95:
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159:, whose definition should not be confused with the differential variational inequality used in Aubin and Cellina (1984).
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This index can be computed for the above examples. For the mechanical impact example, if we differentiate
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as this corresponds to some kind of adhesive.) This can be summarized by the complementarity relationship
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is also the set of non-negative real numbers; this is a differential complementarity problem.
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Higher order Moreau's sweeping process. Mathematical formulation and numerical formulation
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constraints. Examples of such problems include, for example, mechanical impact problems,
1510:. However, if we differentiate once more, we can use the differential equation to give
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Closely associated with DVIs are dynamic/differential complementarity problems: if
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problems for contacting bodies, and dynamic economic and related problems such as
1798:
Avi
Mandelbaum (1989) "Dynamic Complementarity Problems", unpublished manuscript.
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is a closed convex cone, then the variational inequality is equivalent to the
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Differential variational inequalities were first formally introduced by
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446:{\displaystyle {\frac {dx}{dt}}=f(t,x(t),u(t)),\quad x(t_{0})=x_{0}.}
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DVIs are useful for representing models involving both dynamics and
558:{\displaystyle K\ni u(t)\quad \perp \quad F(t,x(t),u(t))\in K^{*}.}
1805:", SIAM Journal on Applied Mathematics, vol. 60, no. 4, 1234–1269.
121:
1779:", Mathematical Programming, vol. 113, no. 2, Series A, 345–424.
1278:, then there is a complementarity relationship between the two:
15:
1394:) = 0 needed in order to locally uniquely identify
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Differential variational inequalities have the form to find
285:{\displaystyle \langle v-u(t),F(t,x(t),u(t))\rangle \geq 0}
1119:{\displaystyle 0\leq y(t)-r\quad \perp \quad N(t)\geq 0.}
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suitable space of solutions (Schwartz' distributions).
1332:{\displaystyle 0\leq v(t)\quad \perp \quad i(t)\geq 0}
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can take on any non-negative value. (We do not allow
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755:is the gravitational acceleration. Note that both
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43:but its sources remain unclear because it lacks
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88:differential variational inequality (DVI)
74:Learn how and when to remove this message
1481:, which does not yet explicitly involve
1168:{\displaystyle K=\{\,z\mid z\geq 0\,\}}
735:is the contact force of the table, and
1777:Differential Variational Inequalities
1129:In the above formulation, we can set
7:
1601:{\displaystyle d^{2}y/dt^{2}=(1/m)}
1212:Ideal diodes in electrical circuits
1693:{\displaystyle d^{2}y/dt^{2}=b(t)}
146:parabolic variational inequalities
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1608:, which does explicitly involve
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578:Consider a rigid ball of radius
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1803:Linear complementarity systems
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329:a closed convex set, where
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1782:Aubin and Cellina (1984)
1775:Pang and Stewart (2008) "
1052:{\displaystyle N(t)<0}
954:. On the other hand, if
190:{\displaystyle u(t)\in K}
142:evolutionary inequalities
1474:{\displaystyle dy/dt(t)}
1175:, so that its dual cone
988:{\displaystyle y(t)-r=0}
130:dynamic traffic networks
104:complementarity problems
100:variational inequalities
29:This article includes a
1785:Differential Inclusions
1201:{\displaystyle K^{*}=K}
462:complementarity problem
134:differential inclusions
58:more precise citations.
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1637:. Furthermore, if
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94:that incorporates
86:In mathematics, a
31:list of references
1819:Dynamical systems
1398:as a function of
1355:{\displaystyle t}
871:{\displaystyle t}
748:{\displaystyle g}
699:{\displaystyle m}
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591:{\displaystyle r}
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50:Please help
42:
1220:voltage is
56:introducing
1770:References
295:for every
197:such that
111:inequality
1572:−
1402:and
1324:≥
1308:⊥
1292:≤
1188:∗
1156:≥
1150:∣
1111:≥
1095:⊥
1088:−
1073:≤
974:−
898:−
843:≥
837:−
650:−
548:∗
540:∈
494:⊥
478:∋
306:∈
277:≥
274:⟩
214:−
208:⟨
182:∈
64:June 2020
1813:Category
1342:for all
858:for all
815:a priori
569:Examples
1390:,
1386:,
1218:reverse
995:, then
878:. If
157:Stewart
52:improve
686:where
144:, and
122:diodes
1372:index
1366:Index
919:then
119:ideal
117:with
90:is a
37:, or
1044:<
904:>
813:are
784:and
155:and
153:Pang
98:and
102:or
1815::
1758:.
1406:.
1114:0.
464::
325:;
148:.
140:,
136:,
124:,
106:.
41:,
33:,
1746:)
1743:t
1740:(
1737:b
1717:)
1714:t
1711:(
1708:N
1688:)
1685:t
1682:(
1679:b
1676:=
1671:2
1667:t
1663:d
1659:/
1655:y
1650:2
1646:d
1625:)
1622:t
1619:(
1616:N
1596:]
1593:)
1590:t
1587:(
1584:N
1581:+
1578:g
1575:m
1569:[
1566:)
1563:m
1559:/
1555:1
1552:(
1549:=
1544:2
1540:t
1536:d
1532:/
1528:y
1523:2
1519:d
1498:)
1495:t
1492:(
1489:N
1469:)
1466:t
1463:(
1460:t
1457:d
1453:/
1449:y
1446:d
1426:)
1423:t
1420:(
1417:y
1404:x
1400:t
1396:u
1392:u
1388:x
1384:t
1382:(
1380:F
1350:t
1327:0
1321:)
1318:t
1315:(
1312:i
1304:)
1301:t
1298:(
1295:v
1289:0
1266:)
1263:t
1260:(
1257:i
1237:)
1234:t
1231:(
1228:v
1196:K
1193:=
1184:K
1163:}
1159:0
1153:z
1147:z
1143:{
1140:=
1137:K
1108:)
1105:t
1102:(
1099:N
1091:r
1085:)
1082:t
1079:(
1076:y
1070:0
1047:0
1041:)
1038:t
1035:(
1032:N
1012:)
1009:t
1006:(
1003:N
983:0
980:=
977:r
971:)
968:t
965:(
962:y
942:0
939:=
936:)
933:t
930:(
927:N
907:0
901:r
895:)
892:t
889:(
886:y
866:t
846:0
840:r
834:)
831:t
828:(
825:y
801:)
798:t
795:(
792:N
772:)
769:t
766:(
763:y
743:g
723:)
720:t
717:(
714:N
694:m
671:)
668:t
665:(
662:N
659:+
656:g
653:m
647:=
639:2
635:t
631:d
626:y
621:2
617:d
610:m
586:r
553:.
544:K
537:)
534:)
531:t
528:(
525:u
522:,
519:)
516:t
513:(
510:x
507:,
504:t
501:(
498:F
490:)
487:t
484:(
481:u
475:K
458:K
441:.
436:0
432:x
428:=
425:)
420:0
416:t
412:(
409:x
405:,
402:)
399:)
396:t
393:(
390:u
387:,
384:)
381:t
378:(
375:x
372:,
369:t
366:(
363:f
360:=
354:t
351:d
346:x
343:d
327:K
323:t
309:K
303:v
280:0
271:)
268:)
265:t
262:(
259:u
256:,
253:)
250:t
247:(
244:x
241:,
238:t
235:(
232:F
229:,
226:)
223:t
220:(
217:u
211:v
185:K
179:)
176:t
173:(
170:u
77:)
71:(
66:)
62:(
48:.
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