Introductory Optimization Dynamics Optimal Control with Economics an
Optimal Control theory has been increasingly used in Economi- and Management Science in the last fifteen years or so. It is now commonplace, even at textbook level. It has been applied to a great many areas of Economics and Management Science, such as Opt
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Introductory Optimization Dynamics Optimal Control with Economics and Management Science Applications
With 85 Figures
Springer-Verlag Berlin Heidelberg GmbH 1984
Dr. Pierre Ninh Van Tu Associate Professor Department of Economics The University of Calgary 2500 University Drive, N. W. Calgary, Alberta T2N 1N4 Canada
ISBN 978-3-540-13305-6 DOI 10.1007/978-3-662-00719-8
ISBN 978-3-662-00719-8 (eBook)
This wor1< is subject to copyright. All rights are reserved, whether the whole or part of materials is concemed, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
©Springer-Verlag Berlin Heidelberg 1984 Originally published by Springer-Verlag Berlin Heidelberg New York in 1984.
Softcover reprint of the hardcover I st edition 1984 The use of registered names, trademarl ~
=
where x is an n-vector and
o
(l~i~ 2 :
variable terminal time
Euler equation is 1-ZX=O
giving
= l4 t 2 +
x*(t)
c t + c2 1
where c 1 and c 2 are arbitrary constants to be determined in each case as follows: (a)
x(O) • 1 "" c 2 x(2)
=1 +
2c 1 + 1
= 10,
i.e. c 1
•
4 •
The solution is
x*(t) (b)
x(O)
= t 2 /4 + 4t + 1 •
= 1 = c2
c 1 is determined by the condition (4), i.e. t •
T, i.e. x(T) • 0
=
tT+c
x* (t)
1
t 2 /4
1 + c1
-
t
+ 1
=
fx •
0, i.e. c
1
2X • 0 at
=-
1 .
Thus
62 (c)
= 1 = c2
x(O)
c 1 is determined by condition (11) i.e.
2, we have (~)t 2 - t + 1
x*(t) T*
6
Example 3. 2. 2. Consider the problem of minimizing the cost of building a road from a given point A(O,l) of a city to an existing highway situated at g(t)
= 11
(put AC
=1
- 2t.
Assuming constant average construction cost AC
for convenience) per kilometre, this amounts to minimizing
J(x)
T
I
1
(1 + X2)~
dt
0
where x(O)
=1
but T and x(T) are both unspecified except that x(T)
must lie on the line g(t)
=
brachistochrone problem.
The Euler equation gives
i.e.
~
= 0,
11 - 2t.
the solution of which is
This is a variable end point
d~ (1 + ;2)-~;
=
0
63 The constants c 1 _and c 2 are determined by .:r(O) = c 2 = 1
f + 0.
[ '(k) - 6) g
This, together with
k = g(k)
determines the behaviour of the system. is given by the {k*,c*) at which constant c (i.e.
g'(k*) (i.e.
a
k
c ..
k. = 0
One solution to this system =
0)., a capital level
6 (or f'(k*)
~
- c,
c.
In order to maintain a
k* must exist such that
l + 6) and in order to maintain a constant k,
~ 0), we must have c*
= g(k*)
(: f(k*) - Ak*).
(See Fig.
3.13a and 3.13b.)
lk
0
k*
f(k)
k
f
0) and C
X
= ac/ax :::
0:
= marginal cost (MC)
rising extraction cost as
exhaustion is nearer. From the definition of x(t), it is clear that
i.e.
x
=
q
with x(O)
The objective functional to be maximized is
where F(x,q ,t)
e -rt rr(q,x) above.
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