# Impact of Goodwill on Consumer Buying through Advertising in a Segmented Market: An Optimal Control Theoretic Approach

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## Abstract

**:**

## 1. Introduction

## 2. Model Formulation

## 3. Optimal Policy and Local Stability Analysis

#### 3.1. Optimal Dynamic Strategy for Finite Time Horizon

#### 3.2. Optimal Dynamic Strategy for Infinite Time Horizon

#### 3.3. Local Stability Analysis in the State–Costate-Phase Plane

## 4. Numerical Illustration and Sensitivity Analysis

#### 4.1. Numerical Illustrations

#### 4.2. Local Sensitivity Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

- Proof of Equations (12)–(15):

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Segment | ${\mathit{a}}_{\mathit{i}}$ | ${\mathit{a}}_{\mathit{i}}$ | ${\mathit{a}}_{1\mathit{i}}$ | ${\mathit{a}}_{2\mathit{i}}$ | ${\mathit{c}}_{\mathit{i}}$ | ${\mathit{\delta}}_{\mathit{i}}$ | ${\mathit{\eta}}_{\mathit{i}}$ | ${\mathit{k}}_{\mathit{i}}$ |
---|---|---|---|---|---|---|---|---|

S1 | 1660 | 0.30 | 2.5 | 2.2 | 20 | 0.01 | 2 | 1.5 |

S2 | 1670 | 0.32 | 2.3 | 2.3 | 21 | 0.01 | 3 | 1.6 |

S3 | 1670 | 0.28 | 2.4 | 2.2 | 23 | 0.01 | 2 | 1.5 |

Parameters | ${\mathit{p}}_{1}$ | ${\mathit{p}}_{2}$ | ${\mathit{p}}_{3}$ | ${\mathit{w}}_{1}$ | ${\mathit{w}}_{2}$ | ${\mathit{w}}_{3}$ | w |
---|---|---|---|---|---|---|---|

${c}_{1}$ | 1 | 0 | 0 | 1 | 0 | 0 | 0.3504 |

${c}_{2}$ | 0 | 1 | 0 | 0 | 1 | 0 | 0.2735 |

${c}_{3}$ | 0 | 0 | 0.9565 | 0 | 0 | 1 | 0.3761 |

${\eta}_{1}$ | −1 | 0 | 0 | −1 | 0 | 0 | −0.3504 |

${\eta}_{2}$ | 0 | −0.5 | 0 | 0 | −1 | 0 | −0.2735 |

${\eta}_{3}$ | 0 | 0 | −1 | 0 | 0 | −1 | −0.3761 |

${a}_{21}$ | 0 | 0 | 0 | 1 | 0 | 0 | 0.3504 |

${a}_{22}$ | 0 | 0 | 0 | 0 | 1 | 0 | 0.2735 |

${a}_{23}$ | 0 | 0 | 0 | 0 | 0 | 1 | 0.3761 |

${\delta}_{1}$ | 0 | 0 | 0 | 0.0645 | 0 | 0 | 0.0236 |

${\delta}_{2}$ | 0 | 0 | 0 | 0 | 0.0673 | 0 | 0.0184 |

${\delta}_{3}$ | 0 | 0 | 0 | 0 | 0 | 0.0655 | 0.0253 |

${\alpha}_{1}$ | 0 | 0 | 0 | 0.3504 | 0 | 0 | 0.3504 |

${\alpha}_{2}$ | 0 | 0 | 0 | 0 | 0.2735 | 0 | 0.2735 |

${\alpha}_{3}$ | 0 | 0 | 0 | 0 | 0 | 0.3761 | 0.3761 |

${k}_{1}$ | 0 | 0 | 0 | −1 | 0 | 0 | 0 |

${k}_{2}$ | 0 | 0 | 0 | 0 | −1 | 0 | 0 |

${k}_{3}$ | 0 | 0 | 0 | 0 | 0 | −1 | 0 |

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**MDPI and ACS Style**

Kumar, P.; Chaudhary, K.; Kumar, V.; Chauhan, S.
Impact of Goodwill on Consumer Buying through Advertising in a Segmented Market: An Optimal Control Theoretic Approach. *Axioms* **2023**, *12*, 223.
https://doi.org/10.3390/axioms12020223

**AMA Style**

Kumar P, Chaudhary K, Kumar V, Chauhan S.
Impact of Goodwill on Consumer Buying through Advertising in a Segmented Market: An Optimal Control Theoretic Approach. *Axioms*. 2023; 12(2):223.
https://doi.org/10.3390/axioms12020223

**Chicago/Turabian Style**

Kumar, Pradeep, Kuldeep Chaudhary, Vijay Kumar, and Sudipa Chauhan.
2023. "Impact of Goodwill on Consumer Buying through Advertising in a Segmented Market: An Optimal Control Theoretic Approach" *Axioms* 12, no. 2: 223.
https://doi.org/10.3390/axioms12020223