By Ya. Z. Tsypkin
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Extra info for Adaptation and Learning in Automatic Systems
48) This difference equation has a trivial solution q = 0, since by the definition of c*, we have VJ(c*) = 0. 4). As it is known, two types of stability are distinguished when all the coordinates of the vector q[n] are smali. One is the local stability, and the other is global stability (for any q[n]). In order to investigate the local stability, the gradient VJ(c* + q) must first be approximated by a linear f h c t i o n and the obtained linear difference equation is then tested for stability.
Jn practical applications of the described methods we are faced with some definite difficulties; the expressions which describe the conditions of stability in small or in large always contain the unknown optimal vector c*. In the classical theory of stability, it is always assumed that the value c* is a priori given, but here such an assumption is not justified, since the iterative algorithms are the tools for finding c*. However, this difficulty can be overcome if we employ the concept of absolute stability which is broadly used by Lurie in control theory.
COMMENTS I . I Much literature devoted to the problem of optimality cannot be surveyed here in a brief and clear form. For our purpose, this is not necessary. Therefore, we shall only mention several basic books. First, we mention the book by Feldbaum (1965) which presents various formulations of the problem of optimality and possible approaches to its solution. In the books by Bellman (1957, 1961) this problem is considered on the basis of his method of dynamic programming, and in the book by Pontryagin e t a / .