Optimal Land Use and Pollution Abatement under Discounting: A Spatiotemporal Analysis with State Constraints

We develop a rigorous spatiotemporal framework that incorporates ecological state constraints and reversible pollution dynamics with fertile land as the sole bounded production input. Using an extended Pontryagin Maximum Principle and generalized Kuhn-Tucker conditions, we characterize the structure of optimal solutions across discounting regimes. Thus, this paper investigates optimal strategies for land use and consumption under varying time discount rates in spatial settings. Under low discounting, the system supports sustainable dynamics, avoiding excessive consumption and enabling full restoration of fertile land. Fertility restoration is local in the heterogeneous case and global in the homogeneous case. In contrast, high discounting leads to boundary behavior, partial degradation, or irreversible loss, depending on critical thresholds. We derive explicit solutions when space is homogeneous. In heterogeneous settings, we construct hybrid solutions where the system transitions from a finite-horizon control problem to a structured long-run regime. Our results provide analytical benchmarks and highlight the pivotal role of time preferences in shaping long-term environmental and economic outcomes. This work contributes to the literature on spatial growth, optimal control with state constraints, and sustainable resource management under ecological limits.