Differential inclusion

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 formula_2. Differential inclusions arise in many situations including differential variational inequalities, projected dynamical systems, Moreau's sweeping process, linear and nonlinear complementarity dynamical systems, discontinuous ordinary differential equations, switching dynamical systems, 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 set-valued function.
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 "blow-up" (formula_4 as formula_5 for a finite formula_6).
Existence theory for differential inclusions with non-convex "F"("t", "x") is an active area of research.
Uniqueness of solutions usually requires other conditions. 
For example, suppose formula_7 satisfies a one-sided 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 formula_10, but allows no discontinuities in the state, i.e. formula_11 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 formula_11.
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 non-smooth dynamical systems (NSDS) analysis, which is used in the "analog" study of switching electrical circuits using idealized component equations (for example using idealized, straight vertical lines for the ) and in the study of certain non-smooth mechanical system such as stick-slip oscillations in systems with dry friction or the dynamics of impact phenomena. Software that solves NSDS systems exists, such as INRIA's Siconos.