Input-output systems whose transmission operators furnish nearly best approximations to a given map

Abstract

Let {S(A):A ∈A}, whereA is a subset of an infinite-dimensional normed linear spaceL, be a class of general nonlinear input-output systems that are governed by operator equations relating the input, state, and output, all of which are in extended spaces. IfQ is a given operator from a specified set ¯D _i, of inputs into the space of outputs ¯H ₀, the problem we consider is to find, for a given ɛ>0, a “parameter”A _ε∈A such that the transmission operatorR(A _ε) ofS(A _ε) furnishes a nearly best (or ɛ-best) approximation toQ from allR(A),A ∈A. Here the “distance” betweenQ andR(A) is defined as the supremum of distances betweenQz andR(A)z taken over allz ∈ ¯D _i. In Theorems 2 through 5 we show that ifS(A) is “normal” (Definition 2),A satisfies some mild requirement andL contains a fundamental sequence, then establishingA _ε∈A reduces to minimizing a certain continuous functional on a compact subset ofR ⁿ, and thus can be carried out by conventional methods. The applications of results are illustrated by the example of a model-matching problem for a nonlinear system, and of optimal tracking.