Separation Conditions, Myopic Maps, and Criteria for UniformApproximation of Input-Output Maps

Abstract

This paper considerably extends recent discrete-time (and discrete-space) results concerning the problem of obtaining criteria under which input-output maps can be uniformly approximated arbitrarily well using a certain structure consisting of a not-necessarily linear dynamic part followed by a nonlinear memoryless section that may contain sigmoids or radial basis functions, etc. In those results certain separation conditions of the kind associated with the Stone-Weierstrass theorem play a prominent role and emerge as criteria for approximation—not just sufficient conditions under which an approximation exists. Here we give corresponding results for a much larger set of maps of interest. More specifically, corresponding results are given for an important family of continuous-time and continuous space systems. As an example, it is shown that a large class of continuous-space myopic systems with continuous inputs can be uniformly approximated arbitrarily well using just a bank of shift operators followed by a nonlinear memoryless section. This directs attention to approximants for such systems of a very different type than those discussed earlier in the literature. In another example, a new result is given concerning the uniform approximation of a large class of myopic continuous-time systems with inputs that may have discontinuities.