Nonlinear Programming and Discrete-Time Optimal Control

Part of the International Series in Operations Research & Management Science book series (ISOR, volume 135)


The primary intent of this chapter is to introduce the reader to the theoretical foundations of nonlinear programming as well as the theoretical foundations of deterministic discrete-time optimal control. In fact, deterministic discrete-time optimal control problems, as we shall see, are actually nonlinear mathematical programs with a very particular type of structure. In a later chapter, we will also discover that deterministic continuous-time optimal control problems are specific instances of mathematical programs in topological vector spaces. Consequently, it is imperative for the student of optimal control to have a command of the foundations of nonlinear programming. Particularly important are the notions of local and global optimality in mathematical programming, the Kuhn-Tucker necessary conditions for optimality in nonlinear programming, and the role played by convexity in making necessary conditions sufficient. Readers already comfortable with finite-dimensional nonlinear programming may wish to go immediately to Section 2.9. We do caution, however, that subsequent chapters of this book assume substantial familiarity with finite-dimensional nonlinear programming, so that an overestimate of one’s nonlinear programming knowledge can be very detrimental to ultimately obtaining a deep understanding of optimal control theory and differential games.


Mathematical Program Nonlinear Programming Inequality Constraint Optimal Control Theory Unconstrained Minimum 
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List of References Cited and Additional Reading

  1. Bazaraa, M., H. Sherali, and C. Shetty (1993). Nonlinear Programming: Theory and Algorithms. New York: John Wiley.Google Scholar
  2. Canon, M., C. Cullum, and E. Polak (1970). Theory of Optimal Control and Mathematical Programming. New York: McGraw-Hill.Google Scholar
  3. Luenberger, D. G. (1984). Linear and Nonlinear Programming. Reading, MA: Addison-Wesley.Google Scholar
  4. Mangasarian, O. (1969). Nonlinear Programming. New York: McGraw-Hill.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Dept. Industrial & Manufacturing EngineeringPennsylvania State UniversityUniversity ParkUSA

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