Book contents
- Frontmatter
- Contents
- Foreword by Eugene Silberberg
- Preface
- 1 Essential Elements of Continuous Time Dynamic Optimization
- 2 Necessary Conditions for a Simplified Control Problem
- 3 Concavity and Sufficiency in Optimal Control Problems
- 4 The Maximum Principle and Economic Interpretations
- 5 Linear Optimal Control Problems
- 6 Necessary and Sufficient Conditions for a General Class of Control Problems
- 7 Necessary and Sufficient Conditions for Isoperimetric Problems
- 8 Economic Characterization of Reciprocal Isoperimetric Problems
- 9 The Dynamic Envelope Theorem and Economic Interpretations
- 10 The Dynamic Envelope Theorem and Transversality Conditions
- 11 Comparative Dynamics via Envelope Methods
- 12 Discounting, Current Values, and Time Consistency
- 13 Local Stability and Phase Portraits of Autonomous Differential Equations
- 14 Necessary and Sufficient Conditions for Infinite Horizon Control Problems
- 15 The Neoclassical Optimal Economic Growth Model
- 16 A Dynamic Limit Pricing Model of the Firm
- 17 The Adjustment Cost Model of the Firm
- 18 Qualitative Properties of Infinite Horizon Optimal Control Problems with One State Variable and One Control Variable
- 19 Dynamic Programming and the Hamilton-Jacobi-Bellman Equation
- 20 Intertemporal Duality in the Adjustment Cost Model of the Firm
- Index
- References
5 - Linear Optimal Control Problems
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Foreword by Eugene Silberberg
- Preface
- 1 Essential Elements of Continuous Time Dynamic Optimization
- 2 Necessary Conditions for a Simplified Control Problem
- 3 Concavity and Sufficiency in Optimal Control Problems
- 4 The Maximum Principle and Economic Interpretations
- 5 Linear Optimal Control Problems
- 6 Necessary and Sufficient Conditions for a General Class of Control Problems
- 7 Necessary and Sufficient Conditions for Isoperimetric Problems
- 8 Economic Characterization of Reciprocal Isoperimetric Problems
- 9 The Dynamic Envelope Theorem and Economic Interpretations
- 10 The Dynamic Envelope Theorem and Transversality Conditions
- 11 Comparative Dynamics via Envelope Methods
- 12 Discounting, Current Values, and Time Consistency
- 13 Local Stability and Phase Portraits of Autonomous Differential Equations
- 14 Necessary and Sufficient Conditions for Infinite Horizon Control Problems
- 15 The Neoclassical Optimal Economic Growth Model
- 16 A Dynamic Limit Pricing Model of the Firm
- 17 The Adjustment Cost Model of the Firm
- 18 Qualitative Properties of Infinite Horizon Optimal Control Problems with One State Variable and One Control Variable
- 19 Dynamic Programming and the Hamilton-Jacobi-Bellman Equation
- 20 Intertemporal Duality in the Adjustment Cost Model of the Firm
- Index
- References
Summary
We now turn to the examination of optimal control problems that are linear in the control variables. A prominent feature of this class of problems is that the optimal control often turns out to be a piecewise continuous function of time. Recall that in Chapter 4, we defined an admissible pair of curves (x(t), u(t)) by allowing the control vector to be a piecewise continuous function of time and the state vector to be a piecewise smooth function of time. Though we allowed for this possibility in the theorems of Chapter 4, we did not solve or confront an optimal control problem whose solution exhibited these properties. You may recall, however, that in Example 4.6, where we solved the ubiquitous inventory accumulation problem subject to a nonnegativity constraint on the production rate, the optimal production rate was a continuous but not a differentiable function of time, that is, it was a piecewise smooth function of time. There are two reasons why the optimal production rate turned out to be a piecewise smooth function of time, namely, (i) the nonnegativity constraint on the production rate and (ii) the assumption of a “long” production period. One important lesson from this example, therefore, is that once an inequality constraint is imposed on the control variable, the differentiability of an optimal control function with respect to time may not hold.
- Type
- Chapter
- Information
- Foundations of Dynamic Economic AnalysisOptimal Control Theory and Applications, pp. 122 - 148Publisher: Cambridge University PressPrint publication year: 2005