论文标题
表征Filippov代表地图和Clarke细分
Characterization of Filippov representable maps and Clarke subdifferentials
论文作者
论文摘要
普通微分方程$ \ dot {x}（t）= f（x（t）），\; t \ geq 0 $，对于$ f $可测量，不足以保证解决方案的存在。为了解决这个问题，我们可以通过用其filippov正则化$ f_ {f} $替换功能$ f $来放松问题，并考虑在f_ {f}（x（t））$中的微分包含$ \ dot $ \ dot {x}（x}（t）\，该$始终具有解决方案。有趣的是，当可以作为（单值，可测量）函数的Filippov正则化获得设置值$φ$时，这很有趣。在这项工作中，我们给出了此类设置值地图的全面表征，特此称为Filippov代表。这种特征还提供了对Lipschitz功能的Clarke细分的那些地图的优雅描述。
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 $Φ$ 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.
