Introduction

The control of fluid motions for the purpose of achieving some desired objective is crucial to many technological applications. In the past, these control problems have been addressed either through expensive experimental processes or through the introduction of significant simplifications into the analyses used in the development of control mechanisms. Only recently have flow control problems been addressed, by scientists and mathematicians, in a systematic, rigorous manner. This interest is quickly expanding so that, at this time, flow control is becoming a very active and successful area of inquiry. For example, recent publications, e.g., [1] [28], provide analyses of various aspects of flow control problems, and include one or more of the following components:

• the construction of mathematical models, invoking minimal assumptions about the physical phenomena;

• the analysis of the mathematical models to answer questions about the existence and regularity of solutions and to derive necessary conditions that optimal controls and states must satisfy;

• the construction and analysis of discretization methods for determining approximate solutions of the optimal control problems, and the rigorous derivation of error estimates; and

• the development of computer codes implementing discretization algorithms, both for the purpose of showing the efficacy of these methods, and also to solve problems of practical interest.

An optimal control or optimization problem is composed of two ingredients: a desired objective and control mechanisms that are used to (hopefully) achieve the desired objective. In a mathematical description of such problems, the desired objective is usually expressed in terms of the extremization of a functional depending on the state of the system, and possibly also on the control mechanisms.