Differential inclusion
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In mathematics, differential inclusions are a generalization of the concept of ordinary differential equation of the form
where F is a multivalued map, i.e. F(t, x) is a set rather than a single point in . Differential inclusions arise in many situations including differential variational inequalities, projected dynamical systems, dynamic Coulomb friction problems and fuzzy set arithmetic.
For example, the basic rule for Coulomb friction is that the friction force has magnitude μN in the direction opposite to the direction of slip, where N is the normal force and μ is a constant (the friction coefficient). However, if the slip is zero, the friction force can be any force in the correct plane with magnitude smaller than or equal to μN Thus, writing the friction force as a function of position and velocity leads to a setvalued function.
Contents
Theory
Existence theory usually assumes that F(t, x) is an upper hemicontinuous function of x, measurable in t, and that F(t, x) is a closed, convex set for all t and x. Existence of solutions for the initial value problem
for a sufficiently small time interval [t_{0}, t_{0} + ε), ε > 0 then follows. Global existence can be shown provided F does not allow "blowup" ( as for a finite ).
Existence theory for differential inclusions with nonconvex F(t, x) is an active area of research.
Uniqueness of solutions usually requires other conditions. For example, suppose satisfies a onesided Lipschitz condition:
for some C for all x_{1} and x_{2}. Then the initial value problem
has a unique solution.
This is closely related to the theory of maximal monotone operators, as developed by Minty and Haïm Brezis.
Filippov's theory only allows for discontinuities in the derivative , but allows no discontinuities in the state, i.e. need be continuous. Schatzman and later Moreau (who gave it the currently accepted name) extended the notion to measure differential inclusion (MDI) in which the inclusion is evaluated by taking the limit from above for .^{[1]}
Applications
Differential inclusions can be used to understand and suitably interpret discontinuous ordinary differential equations, such as arise for Coulomb friction in mechanical systems and ideal switches in power electronics. An important contribution has been made by A. F. Filippov, who studied regularizations of discontinuous equations. Further the technique of regularization was used by N.N. Krasovskii in the theory of differential games.
Differential inclusions are also found at the foundation of nonsmooth dynamical systems (NSDS) analysis,^{[2]} which is used in the analog study of switching electrical circuits using idealized component equations (for example using idealized, straight vertical lines for the sharply exponential forward and breakdown conduction regions of a diode characteristic)^{[3]} and in the study of certain nonsmooth mechanical system such as stickslip oscillations in systems with dry friction or the dynamics of impact phenomena.^{[4]} Software that solves NSDS systems exists, such as INRIA's Siconos.
See also
 Stiffness, which affects ODEs/DAEs for functions with "sharp turns" and which affects numerical convergence
References
 ^ David E. Stewart (2011). Dynamics with Inequalities: Impacts and Hard Constraints. SIAM. p. 125. ISBN 9781611970708.
 ^ Markus Kunze (2000). NonSmooth Dynamical Systems. Springer Science & Business Media. ISBN 9783540679936.
 ^ Vincent Acary; Olivier Bonnefon; Bernard Brogliato (2010). Nonsmooth Modeling and Simulation for Switched Circuits. Springer Science & Business Media. pp. 3–4. ISBN 9789048196814.
 ^ Remco I. Leine; Hendrik Nijmeijer (2013). Dynamics and Bifurcations of NonSmooth Mechanical Systems. Springer Science & Business Media. p. V (preface). ISBN 9783540443988.
 Aubin, JeanPierre; Cellina, Arrigo (1984). Differential Inclusions, SetValued Maps And Viability Theory. Grundl. der Math. Wiss. 264. Berlin: Springer. ISBN 9783540131052.
 Aubin, JeanPierre; Frankowska, Hélène (1990). SetValued Analysis. Birkhäuser. ISBN 9780817648473.
 Deimling, Klaus (1992). Multivalued Differential Equations. Walter de Gruyter. ISBN 9783110132120.
 Andres, J.; Górniewicz, Lech (2003). Topological Fixed Point Principles for Boundary Value Problems. Springer. ISBN 9789048163182.
 Filippov, A.F. (1988). Differential equations with discontinuous righthand sides. Kluwer Academic Publishers Group. ISBN 902772699X.