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Erschienen in:
2013 | OriginalPaper | Buchkapitel
26. Additional exercises for Part IV
verfasst von : Francis Clarke
Erschienen in: Functional Analysis, Calculus of Variations and Optimal Control
Verlag: Springer London
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Abstract
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: 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.
$$J(x ,u)\: = \:\int_{ 0}^{\,T} e^{-\delta t}\big(\, x(t)+u(t)\big)\, dt\,. $$