# A tutorial on the deterministic impulse control maximum principle

## Necessary and sufficient optimality conditions

M. Chahim, R.F. Hartl, P.M. Kort

Research output: Contribution to journalArticleScientificpeer-review

### 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.
Original language English 18-26 European Journal of Operational Research 219 1 https://doi.org/10.1016/j.ejor.2011.12.035 Published - 2012

### Fingerprint

Impulse Control
Necessary and Sufficient Optimality Conditions
Maximum principle
Maximum Principle
Necessary Optimality Conditions
Sufficiency
Optimal Solution
Necessary Conditions of Optimality
Concave function
Concavity
Costs
Multiple Solutions
Violate
Cost Function
Optimal Control Problem
Jump
Cost functions
Tutorial
Impulse control
Optimality conditions

### Cite this

@article{1affd6421b8b402eb5eb0e917254f793,
title = "A tutorial on the deterministic impulse control maximum principle: Necessary and sufficient optimality conditions",
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.",
author = "M. Chahim and R.F. Hartl and P.M. Kort",
note = "Appeared earlier as CentER Discussion Paper 2011-133",
year = "2012",
doi = "10.1016/j.ejor.2011.12.035",
language = "English",
volume = "219",
pages = "18--26",
journal = "European Journal of Operational Research",
issn = "0377-2217",
publisher = "Elsevier Science BV",
number = "1",

}

In: European Journal of Operational Research, Vol. 219, No. 1, 2012, p. 18-26.

Research output: Contribution to journalArticleScientificpeer-review

TY - JOUR

T1 - A tutorial on the deterministic impulse control maximum principle

T2 - Necessary and sufficient optimality conditions

AU - Chahim, M.

AU - Hartl, R.F.

AU - Kort, P.M.

N1 - Appeared earlier as CentER Discussion Paper 2011-133

PY - 2012

Y1 - 2012

N2 - 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.

AB - 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.

U2 - 10.1016/j.ejor.2011.12.035

DO - 10.1016/j.ejor.2011.12.035

M3 - Article

VL - 219

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EP - 26

JO - European Journal of Operational Research

JF - European Journal of Operational Research

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ER -