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

We develop an optimal control framework for infinite-dimensional systems with inequality state constraints, extending the Pontryagin Maximum Principle to diffusiondriven dynamics with bounded states. The resulting conditions feature Radon-measure multipliers that characterize boundary behavior in distributed environments. As an illustration, 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 diffusion and bounded state variables.