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


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


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.

Suggested Citation

  • Chahim, Mohammed & Hartl, Richard F. & Kort, Peter M., 2012. "A tutorial on the deterministic Impulse Control Maximum Principle: Necessary and sufficient optimality conditions," European Journal of Operational Research, Elsevier, vol. 219(1), pages 18-26.
  • Handle: RePEc:eee:ejores:v:219:y:2012:i:1:p:18-26 DOI: 10.1016/j.ejor.2011.12.035

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    References listed on IDEAS

    1. Dale Jorgenson, 1967. "The Theory of Investment Behavior," NBER Chapters,in: Determinants of Investment Behavior, pages 129-175 National Bureau of Economic Research, Inc.
    2. Katrin Erdlenbruch & Alain Jean-Marie & Michel Moreaux & Mabel Tidball, 2013. "Optimality of impulse harvesting policies," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 52(2), pages 429-459, March.
    3. 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.
    4. 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.
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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. M. Chahim & D. Grass & R. F. Hartl & P. M. Kort, 2017. "Product innovation with lumpy investment," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 25(1), pages 159-182, March.
    3. Grames, Johanna & Grass, Dieter & Kort, Peter M. & Fürnkranz-Prskawetz, Alexia, 2017. "Optimal investment and location decisions of a firm in a flood risk area using Impulse Control Theory," ECON WPS - Vienna University of Technology Working Papers in Economic Theory and Policy 01/2017, Vienna University of Technology, Institute for Mathematical Methods in Economics, Research Group Economics (ECON).
    4. Korn, Ralf & Melnyk, Yaroslav & Seifried, Frank Thomas, 2017. "Stochastic impulse control with regime-switching dynamics," European Journal of Operational Research, Elsevier, vol. 260(3), pages 1024-1042.
    5. Chahim, M. & Brekelmans, R.C.M. & den Hertog, D. & 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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