Additional exercises for Part IV

  • Francis Clarke
Part of the Graduate Texts in Mathematics book series (GTM, volume 264)


This chapter consists of a collection of exercises on optimal control, the first of which is the following: Optimal control problems which arise in economics or finance often feature a discount rate δ. This parameter is used to express the present value of future revenues or expenses. We introduce this consideration in the problem described on p. 445, by modifying the cost functional (but nothing else) as follows:
$$J(x ,u)\: = \:\int_{ 0}^{\,T} e^{-\delta t}\big(\, x(t)+u(t)\big)\, dt\,. $$
For small δ>0, it can be shown that the optimal process has the same general turnpike nature as before, but with a different turnpike value (depending on δ) for the state. Determine that value.


Functional Analysis Control Problem System Theory Optimal Process Discount Rate 
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  1. [13]
    F. H. Clarke. Optimization and Nonsmooth Analysis. Wiley-Interscience, New York, 1983. Republished as vol. 5 of Classics in Applied Mathematics, SIAM, 1990. Google Scholar

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© Springer-Verlag London 2013

Authors and Affiliations

  • Francis Clarke
    • 1
  1. 1.Institut Camille JordanUniversité Claude Bernard Lyon 1VilleurbanneFrance

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