Course on Pontryagin's Maximum Principle
Instructor
Andrew Lewis
Department of Mathematics and Statistics
Queen's University
andrew@mast.queensu.ca
Lectures: May 9-16 2006.
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Some version of these notes were distributed to students at the beginning of
the course. However, they have been significantly revised since the original
distribution. If you have a version of these notes with a date earlier than
16/05/2006, please download a later version. Earlier versions had aggredious
errors. I will continue to make modifications to these until I stop doing so
at some undetermined time.
[Course notes]
Course description
This is a short course (maybe 20 hours) offered to graduate students at
Universitat Politècnica de Catalunya.
The intent of the course is to introduce students to Pontryagin's Maximum
Principle. The structure of the course is as follows.
- We begin by covering a few topics from the calculus of variations: the
Euler-Lagrange equations, the Legendre condition, and the Weierstrass excess
function.
- An attempt will be made to draw a line from the calculus of variations
to the Maximum Principle through the so-called Skinner-Rusk formulation of
the calculus of variations.
- We clearly define control systems, classes of trajectories, and the
various optimal control problems we consider. We then clearly state the
Maximum Principle.
- To prove the Maximum Principle, we take a big detour through topics such
as control variations and the relationship between these and the reachable
set. By doing this we will indicate how the Maximum Principle is not just
useful in optimal control, but in control theory in general.
- We prove the Maximum Principle.
- We discuss some aspects of the Maximum Principle that are really only
meaningful after one understands the proof.
- The Maximum Principle will be used to say some things about linear
quadratic optimal control and linear time-optimal control.
Prerequisite: Basic real analysis and differential equations are the
only things that will really be essential prerequisites. However,
familiarity with advanced topics in differential equations might be helpful
at times.
Course material
The course will follow the course notes. Students might also benefit by
reading from the following books.
The Mathematical Theory of Optimal Processes
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko
Gordon & Breach
1986.
This is the original reference (first published in Russian in 1961), and has
in many ways not been excelled in its presentation of the Maximum Principle.
Foundations of Optimal Control Theory
E. B. Lee, L. Markus
John Wiley and Sons
1967.
This is an excellent text on optimal control in general. As concerns the
Maximum Principle, it provides some alternative treatments to the original
presentation by Pontryagin, et al. In particular, it isolates the
relationship between the Maximum Principle and the boundary of the reachable
set. We shall have a great deal to say about this in our presentation.
Optimal Control. An Introduction to the Theory and its Applications
M. Athans and P. L. Falb
McGraw-Hill
1966.
This text is written for an engineering audience. It therefore sacrifices
some rigour to gain intuition. Nonetheless, it can be a valuable resource
in the early stages of trying to understand the Maximum Principle. And it
is also a good place to see how the Maximum Principle can be applied.