A new mathematical discipline has emerged from the bustling scientific activity of the last fifteen years, the theory of control processes. Its mathematical horizons are unlimited and its applications increase in range and importance with each passing year. Consequently, it is reasonable to believe that introductory courses in control theory are essential for training the modern graduate student in pure and applied mathematics, engineering, mathematical physics, economics, biology, operations research, and related fields.The extremely rapid growth of the theory, associated intimately with the continuing trend toward automation, makes it imperative that courses of this nature rest upon a broad basis. In this first volume, we wish to cover the fundamentals of the calculus of variations, dynamic programming, discrete control processes, use of the digital computer, and functional analysis.