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Optimal Control of a Bioeconomic Crop-Energy System with Energy Reinvestment

Othman Cherkaoui Dekkaki

TL;DR

The paper addresses the dynamic allocation of crop residues between bioenergy and soil restoration, introducing a circular reinvestment channel where a fraction $\theta$ of accumulated energy feeds back to soil fertility. A three-state, linear-affine continuous-time OCP with a single control $u(t)\in[0,1]$ is analyzed using the Pontryagin Maximum Principle in current-value form, yielding a bang–interior–bang optimal structure and explicit interior control when feasible. Direct numerical optimization confirms a mid-horizon interior phase (quasi-turnpike behavior) and shows how planning horizon and reinvestment efficiency shape switching times and interior duration; comparisons with $\theta=0$ illustrate the strategic advantage of the reinvestment channel. The study demonstrates that circular bioeconomy principles, operationalized via a residue-energy–soil feedback, can enhance sustainable residue management by enabling early energy extraction without compromising long-term soil health, with practical implications for policy design and incentive schemes.

Abstract

We develop an optimal control model for allocating agricultural crop residues between bioenergy production and soil fertility restoration. The system captures a novel circular feedback: a fraction of cumulative energy output is reinvested into soil productivity, linking energy use with ecological regeneration. The dynamics are governed by a nonlinear three-state system describing soil fertility, residue biomass, and accumulated energy, with a single control representing the proportion of biomass diverted to energy. The objective is to maximize a discounted net benefit that accounts for energy revenue, soil value, and operational costs. We apply the Pontryagin Maximum Principle in current-value form to derive necessary optimality conditions and characterize the structure of optimal controls. Numerical simulations based on direct optimization reveal interior and switching regimes, and show how planning horizon and reinvestment efficiency influence optimal strategies. The results highlight the strategic role of energy reinvestment in achieving sustainable residue management.

Optimal Control of a Bioeconomic Crop-Energy System with Energy Reinvestment

TL;DR

The paper addresses the dynamic allocation of crop residues between bioenergy and soil restoration, introducing a circular reinvestment channel where a fraction of accumulated energy feeds back to soil fertility. A three-state, linear-affine continuous-time OCP with a single control is analyzed using the Pontryagin Maximum Principle in current-value form, yielding a bang–interior–bang optimal structure and explicit interior control when feasible. Direct numerical optimization confirms a mid-horizon interior phase (quasi-turnpike behavior) and shows how planning horizon and reinvestment efficiency shape switching times and interior duration; comparisons with illustrate the strategic advantage of the reinvestment channel. The study demonstrates that circular bioeconomy principles, operationalized via a residue-energy–soil feedback, can enhance sustainable residue management by enabling early energy extraction without compromising long-term soil health, with practical implications for policy design and incentive schemes.

Abstract

We develop an optimal control model for allocating agricultural crop residues between bioenergy production and soil fertility restoration. The system captures a novel circular feedback: a fraction of cumulative energy output is reinvested into soil productivity, linking energy use with ecological regeneration. The dynamics are governed by a nonlinear three-state system describing soil fertility, residue biomass, and accumulated energy, with a single control representing the proportion of biomass diverted to energy. The objective is to maximize a discounted net benefit that accounts for energy revenue, soil value, and operational costs. We apply the Pontryagin Maximum Principle in current-value form to derive necessary optimality conditions and characterize the structure of optimal controls. Numerical simulations based on direct optimization reveal interior and switching regimes, and show how planning horizon and reinvestment efficiency influence optimal strategies. The results highlight the strategic role of energy reinvestment in achieving sustainable residue management.

Paper Structure

This paper contains 12 sections, 3 theorems, 26 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Mathematical Formulation
  3. Statement and Existence of the Optimal Control Problem
  4. Pontryagin Maximum Principle Analysis
  5. Link to application.
  6. Numerical Simulations
  7. Parameter context.
  8. Comparison with the No-Reinvestment Scenario ($\theta = 0$)
  9. Effect of Planning Horizon on the Optimal Strategy
  10. Sensitivity analysis
  11. Policy implications.
  12. Conclusion

Key Result

Lemma 1

Let $u(\cdot)\in[0,1]$, $\alpha,\beta,\gamma,\delta_S,\delta_E>0$, and $\theta\ge 0$. If $S(0),R(0),E(0)\ge 0$, then the solution of eq:soil–eq:energy satisfies $S(t),R(t),E(t)\ge 0$ for all $t\in[0,T]$.

Figures (8)

  • Figure 1: Optimal allocation $u(t)$ (top) and switching function $\phi(t)$ (bottom), cf. Eq. \ref{['eq:switching']}. Zeros of $\phi$ identify candidate switches; on intervals where $\phi(t)=0$ the interior control solves the KKT condition, while $\phi(t){<}0$ (resp. $>{}0$) pushes the optimum to the lower (resp. upper) bound. Parameters as in Table \ref{['Tab.param']}.
  • Figure 2: States $S(t)$ (soil), $R(t)$ (residue), and $E(t)$ (cumulative energy). The mid-horizon plateau coincides with interior operation in $u(t)$; early/late drifts reflect bang phases. Parameters as in Table \ref{['Tab.param']}.
  • Figure 3: On the left: the adjoint costate trajectories $\lambda_S(t)$, $\lambda_R(t)$, and $\lambda_E(t)$ obtained via JuMP for the reinvestment model. On the right: the corresponding current-value pseudo-costates $e^{\delta t} \lambda_S(t)$, $e^{\delta t} \lambda_R(t)$, and $e^{\delta t} \lambda_E(t)$. The decaying profiles reflect the diminishing marginal value of state variables under discounting and confirm consistency with the Pontryagin Maximum Principle.
  • Figure 4: No-reinvestment case ($\theta{=}0$): optimal allocation $u(t)$ and switching function $\phi(t){=}\partial H/\partial u$. Zeros of $\phi$ indicate switching times; $\phi{<}0\Rightarrow u{=}0$ (all residues to soil), $\phi{>}0\Rightarrow u{=}1$ (all to energy), and $\phi{\approx}0$ corresponds to interior operation $0{<}u{<}1$. Without $E{\rightarrow}S$ reinforcement, the policy stays at $u{=}0$ for most of the horizon.
  • Figure 5: No-reinvestment case ($\theta{=}0$): state trajectories $S$ (soil), $R$ (residue), and $E$ (cumulative energy). Reduced growth in $S$ and $R$ reflects sustained $u{=}0$; $E$ rises only near the terminal window where $\phi$ becomes positive.
  • ...and 3 more figures

Theorems & Definitions (9)

  • Remark 1
  • Lemma 1: Positivity
  • proof
  • Remark 2
  • Remark 3
  • Theorem 1: Existence of Optimal Control and Boundedness of State Trajectories
  • proof
  • Proposition 1: Structure of the Optimal Control
  • proof