DOC

# Objective function

By Beatrice Jackson,2014-11-26 12:18
15 views 0
Objective function

1

Updated 18.02.04

ECON4925 Resource economics, Spring 2004 Olav Bjerkholt:

Lecture notes on the Theory of Non-renewable Resources Appendix 1: Optimal control

The optimal problem and its solution (according to Hammond et al. (2003, Ch. 11)

Consider the problem

t1

max(,,), ftxudtuU ut0

xgtxtutxtx；；(,(),()), () t00

with one of the terminal conditions imposed:

xtx()((i) x(t)=x (ii) (iii) x(t) free 11 111

**Suppose (x(t), u(t)) is an optimal pair for this problem. Then there exists a

continuous function p(t) such that for all t in [t, t]: 01

**The control function u(t) maximizes the Hamiltonian H( t, x(t), u, p(t)) subject

to , i.e. uU

***HtxtuptHtxtutpt(,(),,())(,(),(),()) for all uU

**ptHtxtutpt()(,(),(),())；; x

For each of the three possible terminal conditions (i), (ii) and (iii) there is a

corresponding transversality condition:

(i’) p(t) no condition 1

*pt()0((ii’) (with p(t)=0 if x(t)>x) 1111

(iii’) p(t)=0 1

If the problem is redefined with t free, then all the conditions given above are satisfied 1*on [t, t], and in addition 01

2

******Htxtutpt(,(),(),())01111

3

The optimal control problem for a nonrenewable resource

S = Stock of nonrenewable resource t

R = Depletion of nonrenewable resource t

TObjective ;rt WFSRtedtmax(,,)ttfunction Rt0

Terminal state SSSS( TTTerminal point T fixedT freeT fixedT free

;rtPresent-value HHSRtFSRteR；；;(,,,)(,,)？？ Hamiltonian

CCurrent-value HSRtFSRtR；；;(,,,)(,,)？， Hamiltonian

Equations of H；;motions SSR；; CH;；;r，，S

HHMax C，？0 (=0 for 0) RtHR CH，；？0 (0 for 0)RtR

Transversality ？？(；？0 (with 0 if )SSSS TTTTconditions ，，(；？0 (with 0 if )SS TTTTno condition on T

H0H0TT CCH0H0TT

Report this document

For any questions or suggestions please email
cust-service@docsford.com