Admissible simulation relations, set-valued feedback, and controlled invariance

This work is a continuation of the author’s study of simulation relations between nonlinear input–output systems with disturbances. Previously we derived a criterion under which a “pointwise” simulation condition implies simulation by so-called “admissible” inputs and disturbances (that is, inputs and disturbances that yield time-dependent vector fields satisfying C1 Carathéodory conditions). This criterion included a certain constant-rank assumption. In this paper we use the theory of set-valued mappings and differential inclusions to derive analogous results in which the constant-rank assumption is replaced by a compactness provision that augments the pointwise simulation condition. We illustrate our simulation results by deriving a sufficient condition for achieving global controlled invariance of a (possibly singular) nonlinear system through the use of a set-valued feedback law.