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    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(,(),(),())0；1111

     3

    The optimal control problem for a nonrenewable resource

    S = Stock of nonrenewable resource t

    R = Depletion of nonrenewable resource t

    TObjective ;rt WFSRtedt；max(,,)ttfunction ：Rt0

    System SRSS；;； tt0

    Terminal state SS；SS( TTTerminal point T fixedT freeT fixedT free

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

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

    Equations of ；H？；;motions ；SSR；; C；H;；;r，，；S

    ；HHMax C，？0 (=0 for 0) RtH；R C；H，；？0 (0 for 0)Rt；R

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

     H；0H；0TT CCH；0H；0TT