2020/03/01 by Mira Bivas, Bivas, Mira, Aris Daniilidis +3
Mathematics · #26E25 #28A10 #28A20 #34A60 #58C06 #FOS: Mathematics #Optimization and Control (math.OC) #math.OC #msc:26E25 #msc:28A10 #msc:28A20 #msc:34A60 #msc:58C06
paper · pdf · doi:10.48550/arxiv.2003.00436
arxiv created 2020/03/01 · arxiv updated 2020/03/03
The ordinary differential equation x(t)=f(x(t)), t ≥ 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 Ff and consider the differential inclusion x(t)∈ Ff(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.