Differential inclusion

Remarks

If $F(t, x) = \set{f(t, x)}$ for a function $f$, the solutions are the solutions of the ordinary differential equation $x' = f(t, x)$ in the sense of Carathéodory. Example: for $F(t, x) = [-1, 1]$ on $\R \times \R$, the function $x(t) = \abs{t}$ is a solution, since it is absolutely continuous and $x'(t) = \pm 1$ for every $t \ne 0$. Non-example: $x(t) = 2t$ is not a solution of the same inclusion, because $x'(t) = 2 \notin [-1, 1]$ at every $t$. An equation with a discontinuous right side, such as $x' = -\sgn x$, is often replaced by the inclusion $x' \in -\operatorname{Sgn}(x)$, where $\operatorname{Sgn}(0) = [-1, 1]$; then $x \equiv 0$ is a solution. Local existence of solutions holds when the values $F(t, x)$ are nonempty, compact and convex and $F$ satisfies suitable upper hemicontinuity and measurability conditions.