Differential inclusion

In mathematics, differential inclusions are a generalization of the concept of ordinary differential equation of the form 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 R d {displaystyle scriptstyle {mathbb {R} }^{d}} . 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 set-valued function. 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 [t0, t0 + ε), ε > 0 then follows.Global existence can be shown provided F does not allow 'blow-up' ( ‖ x ( t ) ‖ → ∞ {displaystyle scriptstyle Vert x(t)Vert , o ,infty } as t → t ∗ {displaystyle scriptstyle t, o ,t^{*}} for a finite t ∗ {displaystyle scriptstyle t^{*}} ). 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 F ( t , x ) {displaystyle F(t,x)} satisfies a one-sided Lipschitz condition: for some C for all x1 and x2. Then the initial value problem