Infinite-dimensional optimization problems involve decision variables that are functions defined over continuous time intervals that may extend to infinity. These formulations are especially appropriate for systems described by differential equations when the objective is to maximize a time-dependent cost or utility functional.

The theoretical foundations rely mainly on the Pontryagin Maximum Principle and the Hamilton-Jacobi-Bellman (HJB) equation, which provide necessary and, in some cases, sufficient optimality conditions [9]. In infinite-dimensional settings, HJB theory has been developed for Banach and Hilbert spaces, providing a rigorous basis for continuous-time decision models under uncertainty [10][11].

The performance of numerical methods in infinite dimensions can also be assessed through recent comparative studies of nonlinear optimization algorithms on benchmark functions, including Rosenbrock’s function [12]. Such studies help adapt solvers more effectively to strongly constrained or unstable problems.

The application of optimal control to water management has enabled researchers to go beyond empirical or purely deterministic models long used in water engineering. These approaches integrate renewal dynamics, variable demand, treatment costs, and infrastructural constraints. Classical, often linear, models such as those proposed by Loucks et al. [5] formed the basis of planning policies in many contexts. However, their ability to represent real systems in sufficient operational detail remains limited. More recent studies increasingly incorporate adaptive feedback, uncertainty, and operational constraints, thereby providing frameworks that are closer to operational decision-making.

The complexity of water planning stems from uncertainty in climatic conditions, demand fluctuations, and infrastructure failures. Stochastic and risk-aware optimization frameworks have been developed to incorporate these sources of uncertainty explicitly [6][13].

Many cities in the Global South experience service disruptions and deteriorating water quality, both of which reduce user satisfaction and operator performance. These aspects remain underrepresented in optimal-control models. The

production function r ( u ) = β u 1 + δ u captures diminishing returns from investment

or management effort and is therefore more suitable for systems affected by saturation or marginal inefficiency.

Complementary references in optimal control emphasize viscosity-solution methods, semiconcavity, and practical control design frameworks [14]-[17]. On the water-management side, classic and modern studies have examined performance criteria, reservoir-operation reviews, simulation-optimization, metaheuristics, and predictive-control strategies for urban networks [18]-[25].

This work proposes both a conceptual and applied advance along four axes:

The application of an infinite-dimensional optimization framework to a vulnerable urban water network, subject to interruptions and severe logistical constraints;The integration of a nonlinear production function, better suited to the reality of diminishing returns in disadvantaged urban areas;The explicit formalization of the impacts of service failures (frequency of outages and degraded quality) on the social utility function;The development of a robust simulator to evaluate alternative planning and decision-support scenarios in a context such as Sarh.

Within this framework, rigorous mathematical formalization is essential to describe the system’s dynamic behavior, incorporate structural constraints, and explore optimal trajectories for effort allocation. The next section develops this framework around an infinite-dimensional optimization model.

We consider a dynamic system governed by the differential equation:

where x ( t ) represents the quantity of water available in the network at time t , and u ( t ) denotes the investment or management effort.

In the context of water management, this optimal-control system incorporates climate forecasts, seasonal variations, and projected needs, thereby supporting infrastructure planning aimed at maximizing social benefit. It also provides a framework for cost control, quality monitoring, and long-term sustainability.

The objective is to maximize the functional:

where ρ > 0 is a discount rate ensuring convergence of the integral [9], and α > 0 is the social weight assigned to user satisfaction.

The control variable is subject to the admissibility constraint:

where the admissible set is defined as

Here L 2 is a Hilbert space equipped with the inner product

which guarantees weak compactness and projection properties [9][11]. The state variable x ( t ) is considered in the Banach space C ( [ 0 , ∞ ) ) , suitable for handling discontinuities arising from service interruptions.

The Hamiltonian is defined as:

The associated adjoint system is:

This system provides the necessary conditions for any optimal pair ( x * , u * ) [9]. For the infinite-horizon benchmark, the usual transversality condition is imposed conceptually,

which rules out trajectories that keep positive shadow-value-weighted stock at infinity.

An alternative characterization is obtained via the value function V ( x , t ) , defined as the maximal achievable utility starting from state x at time t . The HJB equation reads:

where V is assumed differentiable and f sufficiently regular [10][11].

Here, V ′ ( x , t ) denotes the partial derivative of V with respect to x , and f ( x , u , t ) = r ( u ) − d ( t ) . This equation links the optimal value at each state to the trade-off between instantaneous utility and system dynamics.

Under the following assumptions:

f ( x , u , t ) is Lipschitz continuous in x and continuous in u , J ( u ) is upper semi-continuous, U ad is closed, bounded, and convex, S ( x , u , t ) is continuous and bounded,

There exists an optimal solution u * ∈ U ad maximizing J ( u ) . This result follows from classical theorems of weak compactness and weak convergence in Hilbert spaces [9].

The town of Sarh is located in southern Chad on the banks of the Chari River. It is the country’s fourth-largest city, after N’Djamena, Moundou, and Abéché. It covers an area of about 3000 hectares and comprises six districts. For the 2025 scenario used in this paper, we retain a population of 202,873 inhabitants.

The calibration combines measured or report-based inputs with scenario assumptions. Population and urban water-access context are taken from official country-data and WASH monitoring portals [3][4]. Per-capita demand is fixed at 50 L/person/day, a planning value commonly used for basic urban-service scenarios in constrained settings; this yields the annual mean demand level of 10143.65 m³/day. The seasonal amplitude 2028.73 and phase shift −3.37 in d ( t ) are scenario parameters chosen to represent moderate intra-annual variability around that mean rather than a direct statistical fit. The initial stock x 0 = 50000   m 3 and the control bound u max = 20000 are engineering scenarios selected to test medium-scale storage and effort limits; they should therefore be read as assumed planning values, not as audited operational measurements.

Water management in Sarh is then formalized through the following stock dynamics:

To capture realistically the diminishing returns associated with management effort in the urban context of Sarh, we specify water production through a nonlinear law.

Unlike the linear form r ( u ) = β u , we adopt a nonlinear formulation that reflects diminishing returns:

This rational form captures the saturation effect: beyond a certain level of investment, marginal gains become negligible, which is consistent with the technical limits of the local hydraulic system.

In the Sarh context, frequent interruptions ( ω ( t ) ) and degraded water quality ( q ( t ) ) directly influence the perceived level of service. We introduce:

ω ( t ) ∈ { 0 , 1 } : effective service indicator (0 = interruption, 1 = service available); q ( t ) ∈ [ 0 , 1 ] : quality coefficient (1 = perfect quality, 0 = severely degraded quality).

The actual service thus becomes:

where

and the updated instantaneous utility is defined by

where c ( t ) is the unit cost per unit effort u ( t ) . This utility function is the core of the system’s instantaneous performance, combining adjusted user satisfaction with the associated operational cost. We can now formulate the complete optimization problem rigorously while accounting for the system’s dynamic, structural, and functional constraints.

We now have the elements needed to formulate the global problem of optimal water management in Sarh as a continuous-time, infinite-horizon optimal-control problem. The objective is to determine an admissible control policy u ( ⋅ ) that maximizes total discounted utility, while respecting the system’s physical dynamics and operational constraints:

under the following constraints:

where γ ( t ) = ω ( t ) ⋅ q ( t ) ∈ [ 0 , 1 ] models quality of service through interruptions and degraded water quality. This formulation makes explicit:

the nonlinear dynamics of production (diminishing returns),the direct effect of service failures on delivered service,the infinite-time integration specific to strategic planning.

The problem thus posed is well defined under reasonable assumptions on the functions d ( t ) , c ( t ) and γ ( t ) (continuity, boundedness, positivity), and admits an optimal solution according to the existence results presented above.

Objective (5) maximizes discounted welfare by combining delivered service and operating cost.Constraint (6) describes the evolution of the water stock over time under demand pressure and diminishing returns from effort.Constraint (7) states that satisfaction cannot exceed either available supply or demand and is reduced by outages or poor-quality water through γ ( t ) .Constraint (8) restricts the control to realistic, measurable effort paths over long horizons.

The infinite-horizon formulation above is the theoretical benchmark of the paper. The numerical experiments below do not solve the full Pontryagin boundary-value problem or the HJB equation on an infinite domain. Instead, they implement a finite-horizon daily myopic policy over T = 365 days. At each day t , the control is selected on a discrete grid { 0 , 100 , ⋯ , u max } by maximizing the one-step payoff computed with the expected service-quality factor:

The stock is then updated one day forward. The one-year experiment is therefore a truncated approximation of the infinite-horizon model with zero terminal continuation value, that is, V T ( x ) = 0 at the end of the simulated year. Conceptually, this finite-horizon truncation is consistent with the infinite-horizon transversality condition stated above, but it should be interpreted as an operational policy approximation rather than as the exact infinite-horizon optimum.

We consider one year ( T = 365 ) with daily time step ( Δ t = 1 day). Parameters:

Population: 202,873 inhabitants;Per-capita demand: 0.05 m³/person/day (50 L/day);Seasonal baseline demand:

Discount rate: ρ = 0.0005 per day (≈0.1825 annually);Satisfaction weight: α = 1 unless stated otherwise;Production with saturation: r ( u ) = β u 1 + δ u with β = 1 , ;Unit cost (baseline): c = 0.0008 per unit of u per day;Control bounds: u ( t ) ∈ [ 0 , u max ] , u max = 20000 ;Initial stock: x ( 0 ) = 50000   m 3 .

In the baseline, service quality is exogenous: γ ( t ) is drawn independently each day from a three-point distribution,

Thus, outages and quality degradation are not directly controlled by contemporaneous investment u ( t ) in the simulations; they enter as external disturbances. In the final experiment below, we instead use γ ( t ) = 0.8 + 0.2 Z t with Z t ~ Beta ( 5 , 2 ) , again independently across days. All stochastic trajectories reported in this paper are generated with a fixed random seed equal to 2025, and the tables and figures correspond to that reproducible seeded run rather than to Monte Carlo averages. Delivered service is

and the stock updates accordingly. The production capacity at the bound is r ( u max ) ≈ 6667   m 3 / day .

With α = 1 and baseline costs, the objective reduces to quality-adjusted volume minus daily production costs. The total discounted utility is

Figure 1 shows that the stock is rapidly depleted in the first days due to sustained demand; thereafter the system operates in a production-limited regime. Interruptions/degraded quality appear as troughs in s ( t ) . Because unit cost is

Figure 1. Baseline dynamics. Top: stock x ( t ) (navy), demand d ( t ) (orange, dashed), effective service s ( t ) (light blue, raw) and 7-day average (dark green); dotted line: r ( u max ) ≈ 6667   m 3 / day . Bottom: optimal control u ( t ) (3-day median). The stock falls quickly and the system becomes production-limited; service dips reflect interruptions/quality degradation.

small, the control frequently operates near u max . This recurrent saturation is mechanically explained by the one-step objective: with the present calibration, the marginal benefit α   E [ γ t ] r ′ ( u ) exceeds the marginal cost c over most of the admissible range, so the policy continues to increase effort until it reaches the bound. The saturation in r ( u ) softens the gains from very large effort but does not offset the low baseline unit cost.

We increase the unit cost by 10% ( c ↦ 1.1 c ). The discounted utility becomes

Trajectories of stock and service remain close to baseline (Figure 2); the small loss in J is consistent with a high weight on service when α = 1 .

Figure 2. Baseline vs. +10% cost. Seven-day averages of x ( t ) and s ( t ) under baseline and +10% cost. Differences are modest because, with α = 1 , service value dominates cost.

We vary the unit cost by factors { 1 , 2 , 5 , 10 , 50 , 100 } of the baseline c = 0.0008 , keeping the same daily myopic policy with expected γ . Table 1 and Figure 3 summarize: (a) total discounted utility J ; (b) average daily control u ¯ ; (c) representative 3-day-median control paths for factors 1, 10, 100. Utility is relatively flat up to 10× and then erodes, with a sharp drop at 100×; average effort remains near the maximum until costs become extreme.

Table 1.Cost sensitivity under daily myopic control: total discounted utility and control statistics.

Figure 3. Cost sensitivity. (a) Total discounted utility vs. cost multiplier (log scale). (b) Average control u ¯ . (c) Illustrative control paths (3-day medians) for factors 1, 10, 100. Performance is robust to moderate cost increases; utility and effort deteriorate only at very high costs.

We finally consider a more realistic user response and quality profile: set α = 0.6 and draw γ ( t ) ∈ [ 0.8 , 1.0 ] from a Beta-based distribution skewed toward high quality (no full outages). With a high unit cost (100× baseline), effort collapses and the system quickly becomes production-limited; despite acceptable quality, the lower satisfaction weight and high cost depress delivered service (Figure 4).

Figure 4. High cost, imperfect satisfaction, realistic quality. Top: x ( t ) , d ( t ) , and s ( t ) (raw and 7-day average) with capacity line r ( u max ) . The stock depletes early and the system becomes production-limited. Bottom: 3-day median control showing a sustained collapse in effort relative to baseline.