Dynamic Optimization by Morton I. Kamien & Nancy L. Schwartz

Author:Morton I. Kamien & Nancy L. Schwartz
Language: eng
Format: epub
Publisher: Dover Publications, Inc.
Published: 2013-04-17T04:00:00+00:00

Then (2) and (5) result from

Conditions (8) can be generated by maximizing H subject to (3); this is an ordinary nonlinear programming problem in u.

Solve

by appending the constraints to the objective with multipliers w1, w2. The Lagrangian for (9) is (see Section A6)

from which we obtain the necessary conditions for a constrained maximum with respect to u:

Conditions (11)–(13) are equivalent to conditions (8) and constitute an alternative statement of the requirement, as will be shown in Exercise 6. (If u*(t) = a, then b − u* > 0, so (12) requires w1 = 0; hence, from (11), fu + λgu + w2 = 0. Since w2 ≥ 0, we have fu + λgu ≤ 0 if u*(t) = a. This is the first instance in (8). One continues similarly for the other two possibilities.)

Example 1. We solved our production planning problem

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