Nonlinear Programming and Discrete-Time Optimal Control
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.
KeywordsMathematical Program Nonlinear Programming Inequality Constraint Optimal Control Theory Unconstrained Minimum
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List of References Cited and Additional Reading
- Bazaraa, M., H. Sherali, and C. Shetty (1993). Nonlinear Programming: Theory and Algorithms. New York: John Wiley.Google Scholar
- Canon, M., C. Cullum, and E. Polak (1970). Theory of Optimal Control and Mathematical Programming. New York: McGraw-Hill.Google Scholar
- Luenberger, D. G. (1984). Linear and Nonlinear Programming. Reading, MA: Addison-Wesley.Google Scholar
- Mangasarian, O. (1969). Nonlinear Programming. New York: McGraw-Hill.Google Scholar