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

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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 demon- strated. Numerical simulations verify the effectiveness of the proposed control strategy. The framework provides theoretical support for ecological restoration in water-limited regions.

Keywords

vegetation–water model / cross-diffusion / Robin boundary condition / optimal boundary control / adjoint equation / Fréchet differentiability

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