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Optimal Contraception Control Problems in a Nonlinear Size-Structured Vermin Model

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Abstract

This paper investigates a size-structured vermin contraception control model with nonlinear fertility and mortality, in which the control variable appears in both the state equation and the boundary condition. First, we show that the model has a unique non-negative solution, which has a separable form. Then, some continuity results are established, which are important for the purpose of optimal control. In order to make the final size of vermin as small as possible or as close as possible to the ideal distribution under the condition of the lowest control cost, we consider the least cost-size problem and the least cost-deviation problem. For each of the control problems, the optimality conditions are obtained via the adjoint system and tangent-normal cones. For the first problem, we show the existence of the optimal strategy by compactness and extremal sequence. However, for the second problem, we derive the existence of an optimal control via Ekeland’s variational principle. Some numerical simulations have been performed to demonstrate the feasibility of the obtained theoretical results. Numerical results also suggest that decreasing the reproductive rate instead of increasing the mortality is an effective way of managing the impact of vermin.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos. 12001341, 12071418).

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Authors and Affiliations

  1. Shanxi University of Finance and Economics, Taiyuan, China

    Rong Liu

  2. Yuncheng University, Yuncheng, China

    Fengqin Zhang

  3. Wilfrid Laurier University, Waterloo, Canada

    Yuming Chen

Authors
  1. Rong Liu
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  2. Fengqin Zhang
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  3. Yuming Chen
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Correspondence to Fengqin Zhang.

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Communicated by Irena Lasiecka.

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Liu, R., Zhang, F. & Chen, Y. Optimal Contraception Control Problems in a Nonlinear Size-Structured Vermin Model. J Optim Theory Appl 199, 1188–1221 (2023). https://doi.org/10.1007/s10957-023-02246-9

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