Functional equations in adaptive processes and random transmission

By imbedding a complex physical process under consideration within an appropriate class of processes and expressing the functional relationships among the members of the class, it is frequently possible to obtain insights into the structure of the original process which would not be feasible through consideration of that process alone. Not only may analytical expressions be obtained, but frequently computational tools are forged which make possible the exploitation of modern digital computing machines. By way of illustration, this paper is devoted to a discussion of recent applications o f the functional equation techniques of dynamic programming and invariant imbedding to the study of some problems arising in the theory of adaptive control processes and that of transmission through random media. Still other applications which have been made in modulation theory, communication theory, and network analysis are briefly sketched.