The ordinary differential equation $\dot{x}(t)=f(x(t)), \; t \geq 0 $, for $f$ measurable, is not sufficiently regular to guarantee existence of solutions. To remedy this we may relax the problem by replacing the function $f$ with its Filippov regularization $F_{f}$ and consider the differential inclusion $\dot{x}(t)\in F_{f}(x(t))$ which always has a solution. It is interesting to know, inversely, when a set-valued map $\Phi$ can be obtained as the Filippov regularization of a (single-valued, measurable) function. In this work we give a full characterization of such set-valued maps, hereby called Filippov representable. This characterization also yields an elegant description of those maps that are Clarke subdifferentials of a Lipschitz function.

中文翻译：

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Filippov 可表示图和 Clarke 次微分的表征

常微分方程 $\dot{x}(t)=f(x(t)), \; t \geq 0 $，对于 $f$ 可测量，不足以保证解的存在性。为了解决这个问题，我们可以通过将函数 $f$ 替换为其 Filippov 正则化 $F_{f}$ 并考虑微分包含 $\dot{x}(t)\in F_{f}(x(t) )$ 总是有一个解决方案。相反，有趣的是，当一个集合值映射 $\Phi$ 可以作为（单值，可测量）函数的 Filippov 正则化获得时。在这项工作中，我们给出了此类集值映射的完整特征，在此称为可表示的 Filippov。这种表征还对那些是 Lipschitz 函数的克拉克次微分的映射产生了优雅的描述。