Optimal Control of Diffusion Systems with State Constraints: Theory and Application to Land Restoration

Benteng Zou (), Carmen Camacho () and Weihua Ruan ()
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Benteng Zou: DEM, Université du Luxembourg
Carmen Camacho: Paris School of Economics
Weihua Ruan: Purdue University Northwest, USA

DEM Discussion Paper Series from Department of Economics at the University of Luxembourg

Abstract: "We develop an optimal control framework for infinite-dimensional systems with in- equality state constraints, extending the Pontryagin Maximum Principle to diffusion- driven dynamics with bounded states. The resulting conditions feature Radon-measure multipliers that characterize boundary behavior in distributed environments. As an illus- tration, we apply the framework to a model of land fertility evolving through reversible pollution and spatial diffusion. We show how discounting shapes optimal consumption, the activation of state constraints, and long-run spatial patterns. In the homogeneous case, explicit solutions identify conditions for full restoration or persistent degradation, while heterogeneous settings generate hybrid finite-horizon and long-run regimes. The framework provides general analytical tools for dynamic optimization problems with dif- fusion and bounded state variables."

Keywords: "Economic growth; Diffusion; Soil Pollution; Optimal Control; Limited re- sources" (search for similar items in EconPapers)
JEL-codes: C61 O44 Q15 Q56 R11 (search for similar items in EconPapers)
Date: 2026
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