Applied Mathematics Colloquium: Dr. Thomas Seidman

Title: Differential equations with discontinuous right hand side

Speaker: Thomas Seidman, UMBC

Abstract: We consider the existence of solutions of equations of the form y’ = f(y).   When f is Lipschitzian one obtains existence (and uniqueness by Picard iteration (CMP).  When f is only continuous one still (at least in the finite dimensional case) obtains existence, but not uniqueness (Peano Theorem).  Filippov, in the context of f piecewise smooth with smooth separating surfaces, obtained existence of a type of generalized solution.  [This notion included the introduction of “sliding modes”, now an essential ingredient  of hybrid control theory.]   
            The present treatment uses a somewhat related notion of generalized solution to obtain existence without imposing any regularity assumption at all on f.  [This discussion is an outcome of collaboration with Mike Jeffrey (Univ.Bristol, UK).]