Infinite-Dimensional Optimization for Sustainable Urban Water Management: Application to the City of Sarh (Chad)

Abstract

Reliable access to drinking water remains a major challenge for sub-Saharan African cities facing structural, economic, and climatic constraints. This article develops an infinite-dimensional optimization framework for urban water management over an infinite time horizon. The model is formulated within optimal control theory, with investment effort, demand, production, and service disruptions represented as continuous-time functions. A nonlinear production law captures diminishing returns, while outages and quality degradation enter directly into the utility function. The framework is applied to the city of Sarh (Chad) using calibrated scenario data consistent with local operating conditions. Numerical experiments are then conducted over a one-year horizon under a reproducible daily myopic policy, viewed as a finite-horizon approximation of the infinite-horizon benchmark. The results identify regimes of stock depletion, effort saturation, and cost sensitivity, and they show how the framework can support long-term planning in resource-constrained urban areas.