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Network-Based Optimal Control of Pollution Growth

Fausto Gozzi, Marta Leocata, Giulia Pucci

TL;DR

This work develops a network-based, time-continuous model for optimal pollution control under transboundary diffusion, where a central planner chooses brown ${I_i(t)}$ and green ${R_i(t)}$ investments to maximize intertemporal utility ${J}$ that trades consumption, pollution, and renewable costs. By reformulating the objective with a decoupling vector ${\alpha}$, the authors reduce the dynamic problem to a static, per-node optimization and derive explicit optimal controls in cases with convex or linear renewable costs; they also provide long-time behavior results and a special case where renewable productivity is absent. The main contributions include the introduction of a finite-node network framework for spatio-temporal pollution, closed-form optimal policies (or solvable nonlinear systems) across different cost structures, and numerical evidence showing how diffusion structure and node heterogeneity shape investment and pollution outcomes. The results have practical implications for regional environmental policy, illustrating how network topology, diffusion, and technology mix influence optimal investments and long-run pollution levels; the framework also opens avenues for extensions to strategic interactions and capital accumulation.

Abstract

This paper studies a model for the optimal control (by a centralized economic agent which we call the planner) of pollution diffusion over time and space. The controls are the investments in production and depollution and the goal is to maximize an intertemporal utility function. The main novelty is the fact that the spatial component has a network structure. Moreover, in such a time-space setting we also analyze the trade-off between the use of green or non-green technologies: this also seems to be a novelty in such a setting. Extending methods of previous papers, we can solve explicitly the problem in the case of linear costs of pollution.

Network-Based Optimal Control of Pollution Growth

TL;DR

This work develops a network-based, time-continuous model for optimal pollution control under transboundary diffusion, where a central planner chooses brown and green investments to maximize intertemporal utility that trades consumption, pollution, and renewable costs. By reformulating the objective with a decoupling vector , the authors reduce the dynamic problem to a static, per-node optimization and derive explicit optimal controls in cases with convex or linear renewable costs; they also provide long-time behavior results and a special case where renewable productivity is absent. The main contributions include the introduction of a finite-node network framework for spatio-temporal pollution, closed-form optimal policies (or solvable nonlinear systems) across different cost structures, and numerical evidence showing how diffusion structure and node heterogeneity shape investment and pollution outcomes. The results have practical implications for regional environmental policy, illustrating how network topology, diffusion, and technology mix influence optimal investments and long-run pollution levels; the framework also opens avenues for extensions to strategic interactions and capital accumulation.

Abstract

This paper studies a model for the optimal control (by a centralized economic agent which we call the planner) of pollution diffusion over time and space. The controls are the investments in production and depollution and the goal is to maximize an intertemporal utility function. The main novelty is the fact that the spatial component has a network structure. Moreover, in such a time-space setting we also analyze the trade-off between the use of green or non-green technologies: this also seems to be a novelty in such a setting. Extending methods of previous papers, we can solve explicitly the problem in the case of linear costs of pollution.
Paper Structure (21 sections, 8 theorems, 66 equations, 5 figures, 2 tables)

This paper contains 21 sections, 8 theorems, 66 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. A model for Pollution on Networks
  3. On the State Equation and the Well-Posedness of the Objective Function
  4. Solution of the Optimal Control Problem
  5. Rewriting the Objective Function
  6. Explicit Solution of the Problem and Optimal Path
  7. Long-Time Behaviour of the Optimal State Trajectory
  8. The Model with $a_i^R(t) \equiv 1 \; \forall i \in {\mathcal{V}}$.
  9. Some Numerical Investigations when the Renewable Cost Function is Quadratic
  10. Calibration of the Parameters
  11. Numerical Experiments
  12. Conclusions and directions of further research
  13. Proofs
  14. Proof of Proposition \ref{['welldef']}
  15. Proof of Proposition \ref{['prop:alpha_bound']}
  16. ...and 6 more sections

Key Result

Proposition 2.6

$J(p,(I,R))$ is well defined for all $p \in {\mathbb {R}}^n_+$ and $(I,R)\in {\mathcal{A}}(p)$, possibly equal to $+\infty$ or $-\infty$ (depending, respectively, on the occurrences $\gamma \in (0,1)$ and $\gamma \in (1,\infty)$).

Figures (5)

  • Figure 1: Impact of renewable investments on optimal investment choices, long-term pollution levels, production, consumption and emissions across nodes. The parameter values specific to this figure are: $L=L^1$, $\delta_i=0.4$, $a_i^I = 5$. In the straight-line scenario $a_i^R = 1$, in the dashed-lines scenario $a_i^R = 2.75$ at the core and $a_i^R = 1$ at the periphery, $\forall \; i \in {\mathcal{V}}$.
  • Figure 2: Comparison of optimal investment choices, long-term pollution levels, production, consumption and emissions across nodes for different choices of the operator $L$. The parameter values specific to this figure are: $a_i^I = 5$, $a_i^R =2.75$, $\forall\; i \in {\mathcal{V}}$. The parameter $\delta$ varies across nodes, namely $\delta_i=0.3$ at the core and $\delta_i=0.5$ at the periphery. The straight lines represent the case $L=L^1$ and the dashed lines the case $L=L^2$.
  • Figure 3: Comparison of optimal investment choices, long-term pollution levels, production, consumption and emissions across nodes for different choices of the operator $L$. The parameter values specific to this figure are: $a_i^I = 5$, $a_i^R =2.75, \text{wind}=0.4$, $\forall \; i \in {\mathcal{V}}$. The parameter $\delta$ varies across nodes, namely $\delta_i=0.3$ at the core and $\delta_i=0.5$ at the periphery. The straight lines represent the case $L=L^1$ and the dashed lines the case $L=L^3$.
  • Figure 4: Comparison of optimal investment choices, long-term pollution levels, production, consumption and emissions across nodes for different choices of the operator $L$. The model parameters are: $a_i^I = 5$, $a_i^R =2.75$, $\forall \; i \in {\mathcal{V}}$. The parameter $\delta$ varies across nodes, namely $\delta_i=0.3$ at the core and $\delta_i=0.5$ at the periphery. The straight lines represent the case $L=L^1$ and the dashed lines the case $L=L^4$.
  • Figure 5: Comparison of optimal investment choices, long-term pollution levels, production, consumption and emissions across nodes. The parameter values specific to this figure are: $a_i^R=2.75$, $\forall \; i \in {\mathcal{V}}$. The parameter $\delta$ varies across nodes, namely $\delta_i=0.3$ at the core and $\delta_i=0.5$ at the periphery. The straight lines represent the case in which $a^I_i = 5$, $\forall \; i \in {\mathcal{V}}$, and the dashed lines the case in which $a^I_i = 6.5$ on the nodes from $7$ to $15$ and $a^I_i = 2.5$ on the other nodes.

Theorems & Definitions (38)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.4
  • Remark 2.5: Time invariant case
  • Proposition 2.6
  • proof
  • Remark 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • ...and 28 more