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Optimal Control Theory with Applications in Economics$
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Thomas A. Weber

Print publication date: 2011

Print ISBN-13: 9780262015738

Published to MIT Press Scholarship Online: August 2013

DOI: 10.7551/mitpress/9780262015738.001.0001

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Optimal Control Theory

Optimal Control Theory

Chapter:
(p.80) (p.81) 3 Optimal Control Theory
Source:
Optimal Control Theory with Applications in Economics
Author(s):

Thomas A. Weber

Publisher:
The MIT Press
DOI:10.7551/mitpress/9780262015738.003.0003

This chapter introduces the notion of a controllable system, which is a system that can be moved using available controls from one state to another. The chapter is organized as follows. Section 3.2 provides a characterization of controllability for linear systems, both with bounded and unbounded sets of feasible controls. Section 3.3 discusses the simplest version of an optimal control problem and presents necessary as well as sufficient conditions for its solution. Section 3.4 considers the general finite-horizon optimal control problem with endpoint constraints and state-control constraints, which can be effectively dealt with using a more general version of the Pontryagin maximum principle (PMP), the proof of which is discussed in Section 3.6. Section 3.5 deals with optimal control problems where the decision horizon T goes to infinity. Section 3.7 discusses the existence of solutions to optimal control problems, notably the Filippov existence theorem. Exercises are included at the end of the chapter.

Keywords:   control systems, linear systems, finite-horizon optimal control, infinite-horizon optimal control, the Pontryagin maximum principle, Filippov existence theorem

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