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Objective function

By Beatrice Jackson,2014-11-26 12:18
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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

    System SRSS;;; tt0

    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

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