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A tutorial on the deterministic Impulse Control Maximum Principle: Necessary and sufficient optimality conditions

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  • Chahim, Mohammed
  • Hartl, Richard F.
  • Kort, Peter M.

Abstract

This paper considers a class of optimal control problems that allows jumps in the state variable. We present the necessary optimality conditions of the Impulse Control Maximum Principle based on the current value formulation. By reviewing the existing impulse control models in the literature, we point out that meaningful problems do not satisfy the sufficiency conditions. In particular, such problems either have a concave cost function, contain a fixed cost, or have a control-state interaction, which have in common that they each violate the concavity hypotheses used in the sufficiency theorem. The implication is that the corresponding problem in principle has multiple solutions that satisfy the necessary optimality conditions. Moreover, we argue that problems with fixed cost do not satisfy the conditions under which the necessary optimality conditions can be applied. However, we design a transformation, which ensures that the application of the Impulse Control Maximum Principle still provides the optimal solution. Finally, we show for the first time that for some existing models in the literature no optimal solution exists.

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Bibliographic Info

Article provided by Elsevier in its journal European Journal of Operational Research.

Volume (Year): 219 (2012)
Issue (Month): 1 ()
Pages: 18-26

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Handle: RePEc:eee:ejores:v:219:y:2012:i:1:p:18-26

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Web page: http://www.elsevier.com/locate/eor

Related research

Keywords: Impulse Control Maximum Principle; Optimal control; Discrete continuous system; State-jumps; Present value formulation;

References

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  1. Erdlenbruch, Katrin & Jean-Marie, Alain & Moreaux, Michel & Tidball, Mabel, 2010. "Optimality of Impulse Harvesting Policies," IDEI Working Papers 603, Institut d'Économie Industrielle (IDEI), Toulouse.
  2. Dale Jorgenson, 1967. "The Theory of Investment Behavior," NBER Chapters, in: Determinants of Investment Behavior, pages 129-188 National Bureau of Economic Research, Inc.
  3. Luhmer, Alfred, 1986. "A continuous time, deterministic, nonstationary model of economic ordering," European Journal of Operational Research, Elsevier, vol. 24(1), pages 123-135, January.
  4. Gaimon, Cheryl, 1986. "An impulsive control approach to deriving the optimal dynamic mix of manual and automatic output," European Journal of Operational Research, Elsevier, vol. 24(3), pages 360-368, March.
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Cited by:
  1. Grass, D. & Chahim, M., 2012. "Numerical Algorithms for Deterministic Impulse Control Models with Applications," Discussion Paper 2012-081, Tilburg University, Center for Economic Research.
  2. Chahim, M. & Grass, D. & Hartl, R.F. & Kort, P.M., 2012. "Product Innovation with Lumpy Investment," Discussion Paper 2012-074, Tilburg University, Center for Economic Research.
  3. Chahim, M. & Brekelmans, R.C.M. & Hertog, D. den & Kort, P.M., 2012. "An Impulse Control Approach to Dike Height Optimization (Revised version of CentER DP 2011-097)," Discussion Paper 2012-079, Tilburg University, Center for Economic Research.

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