Deterministic and stochastic differential inclusions with multiple surfaces of discontinuity

We consider a class of deterministic and stochastic dynamical systems with discontinuous drift f and solutions that are constrained to live in a given closed domain G in Rn according to a constraint vector field D(·) specified on the boundary ∂G of the domain. Specifically, we consider equations of the form φ = θ + η + u, θ̇ (t) ∈ F(φ(t)), a.e. t for u in an appropriate class of functions, where η is the “constraining term” in the Skorokhod problem specified by (G, D) and F is the set-valued upper semicontinuous envelope of f . The case G = Rn (when there is no constraining mechanism) and u is absolutely continuous corresponds to the well known setting of differential R. Atar was partially supported by the Israel Science Foundation (grant 126/02), the NSF (grant DMS-0600206), and the fund for promotion of research at the Technion. A. Budhiraja was partially supported by the ARO (grants W911NF-04-1-0230,W911NF-0-1-0080). K. Ramanan was partially Supported by the NSF (grants DMS-0406191, DMI-0323668-0000000965, DMS-0405343). R. Atar Department of Electrical Engineering, Technion, Haifa 32000, Israel e-mail: [email protected] A. Budhiraja (B) Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, NC 27599, USA e-mail: [email protected] K. Ramanan Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA e-mail: [email protected]