Differential inclusion

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In mathematics, differential inclusions are a generalization of the concept of ordinary differential equation of the form

${\displaystyle {\frac {dx}{dt}}(t)\in F(t,x(t)),}$

where F is a multivalued map, i.e. F(t, x) is a set rather than a single point in ${\displaystyle \mathbb {R} ^{d}}$. 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.^[1]

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.

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 ${\displaystyle \mathbb {R} ^{N}}$ for some ${\displaystyle N\in \mathbb {N} }$ as following way. Let ${\displaystyle n\in \mathbb {N} }$ and ${\displaystyle E\subset \mathbb {R} ^{n\times n}\setminus \{0\}.}$ Our main purpose is to find a ${\displaystyle W_{0}^{1,\infty }(\Omega ,\mathbb {R} ^{n})}$ function ${\displaystyle u}$ satisfying the differential inclusion ${\displaystyle Du\in E}$ a.e. in ${\displaystyle \Omega ,}$ where ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ is an open bounded set.