<p>Cost-effectiveness analysis plays a pivotal role in the optimal control of vegetation systems. This study establishes an optimal control model for the vegetation-water system by introducing three control variables—replanting and clearing, irrigation, and increasing soil accumulation or crust formation—to identify the most cost-effective management strategy. A theoretical analysis of the no-control model is first presented, and then the existence of a solution to the optimal control problem is proved. Using the Lagrange multiplier method, we derive the first-order necessary conditions. Simulation results indicate that smaller values of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_11677_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\epsilon _1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ϵ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_11677_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\epsilon _2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ϵ</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> tend to favor mixed and single control strategies, respectively. In contrast, variations in the overall cost weight <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_11677_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_11677_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="117" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta =\delta _1 = \delta _2 = \delta _3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo>=</mo> <msub> <mi>δ</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>δ</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>δ</mi> <mn>3</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>) exert minimal influence on strategy selection, with the mixed control consistently emerging as the most cost-effective approach. These findings provide theoretical foundation and practical recommendations for vegetation management.</p>

Optimal control and comprehensive cost-effectiveness analysis of vegetation-water models

期刊论文
发表日期：

2025年8月9日

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Hui-Min Wang,
Gui-Quan Sun,
Jian-Chao Zeng

摘要

Cost-effectiveness analysis plays a pivotal role in the optimal control of vegetation systems. This study establishes an optimal control model for the vegetation-water system by introducing three control variables—replanting and clearing, irrigation, and increasing soil accumulation or crust formation—to identify the most cost-effective management strategy. A theoretical analysis of the no-control model is first presented, and then the existence of a solution to the optimal control problem is proved. Using the Lagrange multiplier method, we derive the first-order necessary conditions. Simulation results indicate that smaller values of $\epsilon _1$ $ϵ_{1}$ and $\epsilon _2$ $ϵ_{2}$ tend to favor mixed and single control strategies, respectively. In contrast, variations in the overall cost weight $\delta $ $δ$ ( $\delta =\delta _1 = \delta _2 = \delta _3$ $δ = δ_{1} = δ_{2} = δ_{3}$ ) exert minimal influence on strategy selection, with the mixed control consistently emerging as the most cost-effective approach. These findings provide theoretical foundation and practical recommendations for vegetation management.