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| In mathematics, '''differential inclusions''' are a generalization of the concept of [[ordinary differential equation]] of the form
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| :<math>\frac{dx}{dt}(t)\in F(t,x(t)), </math>
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| where ''F'' is a [[multivalued function|multivalued map]], i.e. ''F''(''t'', ''x'') is a ''set'' rather than a single point in <math>\scriptstyle{\Bbb R}^d</math>. Differential inclusions arise in many situations including [[differential variational inequality|differential variational inequalities]], [[projected dynamical system]]s, dynamic [[Coulomb friction]] problems and [[fuzzy set]] arithmetic.
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| 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 set-valued function.
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| ==Theory==
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| Existence theory usually assumes that ''F''(''t'', ''x'') is an [[hemicontinuous|upper hemicontinuous]] function of ''x'', measurable in ''t'', and that ''F''(''t'', ''x'') is a closed, convex set for all ''t'' and ''x''.
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| Existence of solutions for the initial value problem
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| :<math>\frac{dx}{dt}(t)\in F(t,x(t)), \quad x(t_0)=x_0</math>
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| for a sufficiently small time interval [''t''<sub>0</sub>, ''t''<sub>0</sub> + ''ε''), ''ε'' > 0 then follows.
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| Global existence can be shown provided ''F'' does not allow "blow-up" (<math>\scriptstyle \Vert x(t)\Vert\,\to\,\infty</math> as <math>\scriptstyle t\,\to\, t^*</math> for a finite <math>\scriptstyle t^*</math>).
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| Existence theory for differential inclusions with non-convex ''F''(''t'', ''x'') is an active area of research.
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| Uniqueness of solutions usually requires other conditions.
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| For example, suppose <math>F(t,x)</math> satisfies a [[Lipschitz_continuity#One-sided_Lipschitz|one-sided Lipschitz condition]]:
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| :<math>(x_1-x_2)^T(F(t,x_1)-F(t,x_2))\leq C\Vert x_1-x_2\Vert^2</math>
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| for some ''C'' for all ''x''<sub>1</sub> and ''x''<sub>2</sub>. Then the initial value problem
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| :<math>\frac{dx}{dt}(t)\in F(t,x(t)), \quad x(t_0)=x_0</math>
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| has a unique solution.
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| This is closely related to the theory of '''maximal monotone operators''', as developed by Minty and [[Haïm Brezis]].
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| ==Applications==
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| 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 [[Aleksei Fedorovich Filippov|A. F. Filippov]], who studied regularizations of discontinuous equations. Further the technique of regularization was used by [[Nikolai Nikolaevich Krasovsky|N.N. Krasovskii]] in the theory of [[differential game]]s.
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| ==References==
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| * {{cite book|first1=Jean-Pierre |last1=Aubin |first2=Arrigo |last2=Cellina |title=Differential Inclusions, Set-Valued Maps And Viability Theory |series=Grundl. der Math. Wiss. |volume=264 |publisher=Springer |location=Berlin |year=1984 |isbn=9783540131052}}
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| * {{cite book|first1=Jean-Pierre |last1=Aubin |first2=Helene |last2=Frankowska |title=Set-Valued Analysis |publisher=Birkhäuser |year=1990 |isbn=978-0817648473}}
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| * {{cite book|first1=Klaus |last1=Deimling |title=Multivalued Differential Equations |publisher=Walter de Gruyter |year=1992 |isbn=978-3110132120}}
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| * {{cite book|first1=J. |last1=Andres |first2=Lech |last2=Górniewicz |title=Topological Fixed Point Principles for Boundary Value Problems|publisher=Springer |year=2003 |isbn=978-9048163182}}
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| * {{cite book|first1=A.F. |last1=Filippov |title=Differential equations with discontinuous right-hand sides|publisher=Kluwer Academic Publishers Group |year=1988 |isbn=90-277-2699-X}}
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| [[Category:Dynamical systems]]
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| [[Category:Variational analysis]]
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