On smoothness of solutions to projected differential equations

Abstract

Projected differential equations are known as fundamental mathematical models in economics, for electric circuits, etc. The present paper studies the (higher order) derivability as well as a generalized type of derivability of solutions of such equations when the set involved for projections is prox-regular with smooth boundary.

Dynamical Systems Coupled with Monotone Set-Valued Operators: Formalisms, Applications, Well-Posedness, and Stability

“…Addressing solutions to such systems definitely requires some novel tools and methodology. • Despite of the fact that the systems we deal with are basically nonsmooth, their solutions can be arbitrarily smooth in some cases, as shown in [497] for the PDS in (2.31), depending on the smoothness of bd(K) and of f (x) + g(t). Thus, it is of interest analyze the structures which contribute to regularity of the solution.…”

On Characterizations of Submanifolds via Smoothness of the Distance Function in Hilbert Spaces

“…A first application we already started to explore is concerned with differential inclusions. In [23], we studied regularity properties of solutions to projected differential inclusions which are inherited from the smoothness of the constraints set. We aim to extend these results to dynamical systems associated with moving sets as the well-known Moreau sweeping process, and also to sets with more complex structures, such as stratifiable and tame sets (for the definitions, see, e.g., [24]).…”