22:
1095:
1489:
Brogliato, Bernard; Tanwani, Aneel (2020). "Dynamical
Systems Coupled with Monotone Set-Valued Operators: Formalisms, Applications, Well-Posedness, and Stability". SIAM Review, vol.62, no 1, pp.3-129, available at
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232:, Moreau's sweeping process, linear and nonlinear complementarity dynamical systems, discontinuous ordinary differential equations, switching dynamical systems, and
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Differential inclusions can be used to understand and suitably interpret discontinuous ordinary differential equations, such as arise for
1408:
study of switching electrical circuits using idealized component equations (for example using idealized, straight vertical lines for the
225:
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In differential inclusion, we not only take a set-valued map at the right hand side but also we can take a subset of a
Euclidean space
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820:
1389:, who studied regularizations of discontinuous equations. Further, the technique of regularization was used by
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in mechanical systems and ideal switches in power electronics. An important contribution has been made by
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1471:, which affects ODEs/DAEs for functions with "sharp turns" and which affects numerical convergence
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is a constant (the friction coefficient). However, if the slip is zero, the friction force can be
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For example, the basic rule for
Coulomb friction is that the friction force has magnitude
1090:{\displaystyle (x_{1}-x_{2})^{T}(F(t,x_{1})-F(t,x_{2}))\leq C\Vert x_{1}-x_{2}\Vert ^{2}}
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sharply exponential forward and breakdown conduction regions of a diode characteristic
259:. Thus, writing the friction force as a function of position and velocity leads to a
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Uniqueness of solutions usually requires other conditions. For example, suppose
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233:
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768: > 0 then follows. Global existence can be shown provided
1226:{\displaystyle {\frac {dx}{dt}}(t)\in F(t,x(t)),\quad x(t_{0})=x_{0}}
736:{\displaystyle {\frac {dx}{dt}}(t)\in F(t,x(t)),\quad x(t_{0})=x_{0}}
255:
force in the correct plane with magnitude smaller than or equal to
1429:
1336:(who gave it the currently accepted name) extended the notion to
400:{\displaystyle E\subset \mathbb {R} ^{n\times n}\setminus \{0\}.}
1692:
Topological Fixed Point
Principles for Boundary Value Problems
1609:. Springer Science & Business Media. p. V (preface).
224:. Differential inclusions arise in many situations including
15:
1631:
Differential
Inclusions, Set-Valued Maps and Viability Theory
1428:
phenomena. Software that solves NSDS systems exists, such as
886:
Existence theory for differential inclusions with non-convex
1400:
Differential inclusions are also found at the foundation of
1253:'s theory only allows for discontinuities in the derivative
810:{\displaystyle \scriptstyle \Vert x(t)\Vert \,\to \,\infty }
1633:. Grundl. der Math. Wiss. Vol. 264. Berlin: Springer.
1580:
Vincent Acary; Olivier
Bonnefon; Bernard Brogliato (2010).
462:{\displaystyle W_{0}^{1,\infty }(\Omega ,\mathbb {R} ^{n})}
1711:
Differential equations with discontinuous right-hand sides
1607:
243:
in the direction opposite to the direction of slip, where
1530:. Springer international Publishing Switzerland, 3rd ed.
1443:
is used in differential inclusion a new concept comes as
1503:
1340:(MDI) in which the inclusion is evaluated by taking the
628:. Existence of solutions for the initial value problem
1584:. Springer Science & Business Media. pp. 3–4.
1582:
Nonsmooth
Modeling and Simulation for Switched Circuits
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1528:Nonsmooth Mechanics. Models, Dynamics and Control
161:{\displaystyle {\frac {dx}{dt}}(t)\in F(t,x(t)),}
569:{\displaystyle \Omega \subset \mathbb {R} ^{n}}
8:
1629:Aubin, Jean-Pierre; Cellina, Arrigo (1984).
1078:
1051:
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385:
847:{\displaystyle \scriptstyle t\,\to \,t^{*}}
1605:Remco I. Leine; Hendrik Nijmeijer (2013).
1559:. Springer Science & Business Media.
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1239:This is closely related to the theory of
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66:Learn how and when to remove this message
746:for a sufficiently small time interval [
29:This article includes a list of general
1481:
382:
82:are a generalization of the concept of
1404:(NSDS) analysis, which is used in the
584:Existence theory usually assumes that
489:satisfying the differential inclusion
1690:Andres, J.; Górniewicz, Lech (2003).
226:differential variational inequalities
7:
1713:. Kluwer Academic Publishers Group.
1292:{\displaystyle {\frac {dx}{dt}}(t)}
1673:Multivalued Differential Equations
1118:. Then the initial value problem
876:{\displaystyle \scriptstyle t^{*}}
803:
620:) is a closed, convex set for all
548:
525:
438:
430:
35:it lacks sufficient corresponding
14:
898:) is an active area of research.
344:{\displaystyle n\in \mathbb {N} }
316:{\displaystyle N\in \mathbb {N} }
288:{\displaystyle \mathbb {R} ^{N}}
217:{\displaystyle \mathbb {R} ^{d}}
20:
1449:Atmospheric dispersion modeling
1190:
700:
1412:) and in the study of certain
1360:
1354:
1338:measure differential inclusion
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407:Our main purpose is to find a
195:rather than a single point in
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84:ordinary differential equation
1:
938:one-sided Lipschitz condition
1555:Non-Smooth Dynamical Systems
1445:Fuzzy differential inclusion
1439:In continuous function when
1414:non-smooth mechanical system
1402:non-smooth dynamical systems
1243:, as developed by Minty and
230:projected dynamical systems
1759:
1526:Bernard Brogliato (2016).
1490:hal.inria.fr/hal-02379498.
1241:maximal monotone operators
772:does not allow "blow-up" (
1501:David E. Stewart (2011).
1447:which has application in
1671:Deimling, Klaus (1992).
576:is an open bounded set.
534:{\displaystyle \Omega ,}
247:is the normal force and
1709:Filippov, A.F. (1988).
1418:stick-slip oscillations
1236:has a unique solution.
511:{\displaystyle Du\in E}
80:differential inclusions
50:more precise citations.
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929:{\displaystyle F(t,x)}
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323:as following way. Let
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1675:. Walter de Gruyter.
1551:Markus Kunze (2000).
1505:. SIAM. p. 125.
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1743:Variational analysis
1648:Aubin, Jean-Pierre;
1366:{\displaystyle x(t)}
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1328:need be continuous.
1321:{\displaystyle x(t)}
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1654:Set-Valued Analysis
1424:or the dynamics of
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261:set-valued function
1650:Frankowska, Hélène
1395:differential games
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1738:Dynamical systems
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1591:978-90-481-9681-4
1566:978-3-540-67993-6
1537:978-3-319-28664-8
1512:978-1-61197-070-8
1393:in the theory of
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482:{\displaystyle u}
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1383:Coulomb friction
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1694:. Springer.
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1453:Cybernetics
1245:Haïm Brezis
608:, and that
48:introducing
1732:Categories
1476:References
1332:and later
31:references
1469:Stiffness
1330:Schatzman
1158:∈
1100:for some
1079:‖
1065:−
1052:‖
1046:≤
1015:−
964:−
868:∗
839:∗
830:→
804:∞
800:→
796:‖
781:‖
668:∈
552:⊂
549:Ω
526:Ω
503:∈
469:function
439:Ω
431:∞
383:∖
375:×
362:⊂
334:∈
306:∈
295:for some
234:fuzzy set
126:∈
1652:(1990).
1463:See also
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1251:Filippov
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518:a.e. in
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580:Theory
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175:is a
1715:ISBN
1696:ISBN
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