### 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 property of the underlying model. 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.

Original language | English |
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Place of Publication | Tilburg |

Publisher | Operations research |

Volume | 2011-052 |

Publication status | Published - 2011 |

### Publication series

Name | CentER Discussion Paper |
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Volume | 2011-052 |

### Keywords

- Impulse Control Maximum Principle
- Optimal Control
- discrete continuous system
- state-jumps
- present value formulation

## Fingerprint Dive into the research topics of 'Impulse Control in Operations Research: Necessary Optimality Conditions, Sufficiency and Existence (replaced by CentER DP 2011-133)'. Together they form a unique fingerprint.

## Cite this

Chahim, M., Hartl, R. F., & Kort, P. M. (2011).

*Impulse Control in Operations Research: Necessary Optimality Conditions, Sufficiency and Existence (replaced by CentER DP 2011-133)*. (CentER Discussion Paper; Vol. 2011-052). Operations research.