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AIP conference proceedings

Quantum approach to optimal pollution control in agro-industrial production for environmental protection

D.T. Muhamediyeva, M. H. Raupova, B. N. Samijonov, Donata Mukhtarova, Maksud Yakhyaev

Published Jan 1, 2026• 0 citations
Publisher

Abstract

In this paper, Pontryagin’s Maximum Principle is used to develop a mathematical model for optimal pollution control. It includes the dynamics of spatial pollution distribution, natural decay, nonlinear saturation, and economic mechanisms for emission regulation. The numerical solution is based on a quantum-classical approach that uses finite difference methods to model pollution dynamics and the Variational Quantum Algorithm to determine optimal pollution control strategies. The optimal control distributions, the evolution of the adjoint variables, and heatmaps of pollutant dispersion, as well as pollution dynamics, are shown for the results. These findings can be used to design environmentally sustainable pollution management strategies that support economic development and environmental protection.

DOI
10.1063/5.0318024

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This paper proposes an optimal control model for carbon dioxide (CO₂) absorption processes based on the Pontryagin maximum principle. The flow rate r(t) is selected as the control variable and determined through quantum optimization using the VQE (Variational Quantum Eigensolver) algorithm. The time-dependent dynamic system is fully characterized by state variables (CO₂ concentration x(t), product concentration p(t), and pollutant concentration s(t)) and their corresponding adjoint variables. The model aims to maximize a utility function that includes reducing CO₂ levels, increasing product output, and minimizing pollution. Results are presented with graphical analysis demonstrating the effectiveness of the VQE algorithm in quantum optimization approaches.
The article considers the solution of the optimal control tasks in ecology and environmental protection based on quantum computing. Pollution dynamics are investigated, with natural decay, diffusion and controlled remediation treatments being considered. Optimal control is designed to minimize environmental impact while also minimizing the costs of cleaning up the environment. To solve the problem effectively, a quantum optimization algorithm is proposed. For this, the desired problem is transformed into a quadratic optimization problem. This enables the application of quantum computing to the formulation of optimal policies for environmental regulation. Experiments are performed, a comparison is made between quantum-classical approaches and conventional techniques, and the feasibility of quantum optimization in sustainability issues is debated.
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Optimal Control Theory is a powerful mathematical tool, which has known a rapid development since the 1950s, mainly for engineering applications. More recently, it has become a widely used method to improve process performance in quantum technologies by means of highly efficient control of quantum dynamics. This tutorial aims at providing an introduction to key concepts of optimal control theory which is accessible to physicists and engineers working in quantum control or in related fields. The different mathematical results are introduced intuitively, before being rigorously stated. This tutorial describes modern aspects of optimal control theory, with a particular focus on the Pontryagin Maximum Principle, which is the main tool for determining open-loop control laws without experimental feedback. The different steps to solve an optimal control problem are discussed, before moving on to more advanced topics such as the existence of optimal solutions or the definition of the different types of extremals, namely normal, abnormal, and singular. The tutorial covers various quantum control issues and describes their mathematical formulation suitable for optimal control. The connection between the Pontryagin Maximum Principle and gradient-based optimization algorithms used for high-dimensional quantum systems is described. The optimal solution of different low-dimensional quantum systems is presented in detail, illustrating how the mathematical tools are applied in a practical way.
Optimal Control Theory is a powerful mathematical tool, which has known a rapid development since the 1950s, mainly for engineering applications. More recently, it has become a widely used method to improve process performance in quantum technologies by means of highly efficient control of quantum dynamics. This review aims at providing an introduction to key concepts of optimal control theory which is accessible to physicists and engineers working in quantum control or in related fields. The different mathematical results are introduced intuitively, before being rigorously stated. This review describes modern aspects of optimal control theory, with a particular focus on the Pontryagin Maximum Principle, which is the main tool for determining open-loop control laws without experimental feedback. The different steps to solve an optimal control problem are discussed, before moving on to more advanced topics such as the existence of optimal solutions or the definition of the different types of extremals, namely normal, abnormal, and singular. The review covers various quantum control issues and describes their mathematical formulation suitable for optimal control. The optimal solution of different low-dimensional quantum systems is presented in detail, illustrating how the mathematical tools are applied in a practical way.
Quantum metrology comprises a set of techniques and protocols that utilize quantum features for parameter estimation which can in principle outperform any procedure based on classical physics. We formulate the quantum metrology in terms of an optimal control problem and apply Pontryagin's maximum principle to determine the optimal protocol that maximizes the quantum Fisher information for a given evolution time. As the quantum Fisher information involves a derivative with respect to the parameter which one wants to estimate, we devise an augmented dynamical system that explicitly includes gradients of the quantum Fisher information. The necessary conditions derived from Pontryagin's maximum principle are used to quantify the quality of the numerical solution. The proposed formalism is generalized to problems with control constraints, and can also be used to maximize the classical Fisher information for a chosen measurement.