Regularization by External Variables

Abstract

Regularization was a big topic at the 2016 CRM Intensive Research Program on Advances in Nonsmooth Dynamics. There are many open questions concerning well known kinds of regularization (e.g. by smoothing or hysteresis). Here we propose a framework for an alternative and important kind of regularization: by external variables. Shadowing in one variable Begin with a one-dimensional dynamical system (1)ẋ = −λ + xb(x; λ) where λ = sign(x). with the sign function being ±1 for x ≷ 0 and having the set value (−1, +1) for x = 0. This has an attracting fixed point on the discontinuity, whereẋ = −λ. Define a switch-shadowing system (2)ẋ = −λ + xb(x; λ) ,ẏ = (x − y)/γ , where λ = sign(y) , or a state-shadowing system (3)ẋ = −λ + yb(y; λ) ,ẏ = (x − y)/γ , where λ = sign(x) ,