Optimal boundary control of a vegetation–water model with cross-diffusion

Jiali Chai; Qimin Zhang; Panrun Li; Jiali Chai; Qimin Zhang; Panrun Li

doi:10.3934/era.2026343

Electronic Research Archive

2026, Volume 34, Issue 11: 8026-8068. doi: 10.3934/era.2026343

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Research article

Optimal boundary control of a vegetation–water model with cross-diffusion

Jiali Chai ¹,
Qimin Zhang ^{1
,
,},
Panrun Li ^{2
,
,}

1.
School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China
2.
Sichuan Vocational College of Information Technology, Guangyuan 628040, China

Received: 31 July 2026 Revised: 11 September 2026 Accepted: 15 September 2026 Published: 22 September 2026

To address vegetation degradation in arid and semi-arid ecosystems, existing vegetation–water models neglect the directional transport of soil water induced by root uptake (cross-diffusion) and human regulation at boundaries. This paper formulates a vegetation–water reaction–diffusion model incorporating a cross-diffusion term and a Robin-type boundary condition. First, the existence and uniqueness of weak solutions are proved with a priori estimates. Second, under bilateral pointwise constraints and an integral constraint on the cumulative boundary-regulation effort, the existence of an optimal boundary control is established via a penalty method. The adjoint system is derived using the Lagrange multiplier method, yielding first-order necessary and second-order sufficient optimality conditions, and the twice continuous Fréchet differentiability of the control-to-state mapping is demonstrated. Numerical simulations verify the effectiveness of the proposed control strategy. The framework provides theoretical support for ecological restoration in water-limited regions.
- vegetation–water model,
- cross-diffusion,
- Robin boundary condition,
- optimal boundary control,
- adjoint equation,
- Fréchet differentiability
Citation: Jiali Chai, Qimin Zhang, Panrun Li. Optimal boundary control of a vegetation–water model with cross-diffusion[J]. Electronic Research Archive, 2026, 34(11): 8026-8068. doi: 10.3934/era.2026343

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Abstract

To address vegetation degradation in arid and semi-arid ecosystems, existing vegetation–water models neglect the directional transport of soil water induced by root uptake (cross-diffusion) and human regulation at boundaries. This paper formulates a vegetation–water reaction–diffusion model incorporating a cross-diffusion term and a Robin-type boundary condition. First, the existence and uniqueness of weak solutions are proved with a priori estimates. Second, under bilateral pointwise constraints and an integral constraint on the cumulative boundary-regulation effort, the existence of an optimal boundary control is established via a penalty method. The adjoint system is derived using the Lagrange multiplier method, yielding first-order necessary and second-order sufficient optimality conditions, and the twice continuous Fréchet differentiability of the control-to-state mapping is demonstrated. Numerical simulations verify the effectiveness of the proposed control strategy. The framework provides theoretical support for ecological restoration in water-limited regions.

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