Pontryagin Maximum Principle Research Articles

Dynamic analysis of SPAFIR epidemic model considering pseudoinfection and perturbations

In this paper, the “pseudoinfection phenomenon” and “individual behavioral responses” are taken into account in the SIR epidemic model, and the deterministic and stochastic models are analyzed dynamically. Through the exploration of the deterministic SPAFIR (Susceptible-Pseudoinfected-Alert-Fragile-Infected-Recovered) model, the conditions for the equilibrium points’ existence and stability are determined. With the assistance of the Pontryagin maximum principle, this paper introduces a Hamiltonian function with a penalty term and the optimal control strategy is obtained by comparing the three groups of strategies. The optimal control strategy requires multiple control measures to inhibit the infectious disease spread. Further considering the stochastic SPAFIR model, the existence of the uniqueness of a global positive solution and the existence of a stationary distribution for the stochastic model are proved, and the condition for disease extinction is also verified. Random perturbations of the effective contact rates can slow the pace of spreading the infectious diseases. Results of theoretical analysis and the system’s responsiveness to variations in effective contact rates are verified by numerical simulations, and the outcomes indicate that active prevention education and community encouragement can both inhibit infection spread.

A Bilevel Optimal Control Method and Application to the Hybrid Electric Vehicle

ABSTRACTIn this article we present a new numerical method based on a bilevel decomposition of optimal control problems. A strong connection between the proposed method and the classical indirect multiple shooting method is shown in the regular case, thanks to a link between the Bellman's value function and the costate from the Pontryagin maximum principle. The value functions are needed in our bilevel decomposition but they are generally difficult to compute. We approximate them by neural networks that have a high potential of generalization and that provide an efficient computation of the gradient of the cost function. We apply the proposed method to an industrial problem, consisting of the determination of torque split and gear shift of a hybrid electric vehicle, the objective being the minimization of the fuel consumption on a given representative cycle. Numerical methods and results are discussed, as well as the possible improvements of the proposed approach.

Analysis and optimal control of a fractional-order SEAIR epidemic model with two-strains

This study focuses on the analysis and optimal control of a fractional-order SEAIR epidemic model, which consists of two strains. The proposed model's well-posedness is evaluated by examining its existence, uniqueness, non-negativity, and boundedness. Furthermore, two basic regeneration numbers are computed, and the model's two equilibrium points are the endemic and disease-free equilibriums. Using suitable Lyapunov functions and LaSalle's invariance principle, we conduct a stability analysis to examine the global stability of these steady states. Ultimately, using Pontryagin's Maximum Principle, we created a time-dependent optimal control problem. We evaluated the impact of model parameters on the dynamics of disease transmission and determined the efficacy of control measures using numerical simulations.

Optimization of the Process of Batch Distillation of Binary Mixtures

The method of dynamic optimization, known as maximum principle of Pontryagin, is presented. It is applied to the process of batch rectification of binary mixtures, with aim of solving the problem of minimum time (for given distillate quantity of specified composition) or maximum distillate quantity of specified composition (for given batch time). The optimum profiles of reflux ratio are found for seven different separation tasks, based on semi-rigorous model without holdups. The results obtained for distillate quantity are compared to respective results obtained with constant reflux ratio and constant distillate composition working policies, and also with the results of other authors. The solution algorithms for all the three working policies are given. Computer implementation of the algorithms is performed in programming language FORTRAN.

Shape Optimization and Experimental Investigation of Glue-Laminated Timber Beams.

This study investigated the optimal shape of glue-laminated timber beams using an analytical model of a slender beam, taking into account the anisotropy of its strength properties as well as boundary conditions at the oblique bottom face of the beam. A control theory problem was formulated in order to optimize the shape of the modeled beam. Two optimization tasks were considered: minimizing material usage (Vmin) for a fixed load-carrying capacity (LCC) of the beam and maximizing load-bearing capacity (Qmax) for a given volume of the beam. The optimal solution was found using Pontryagin's maximum principle (PMP). Optimal shapes were determined using Dircol v. 2.1 software and then adjusted according to a 3D finite element analysis (FEA) performed in Abaqus. The final shapes obtained through this procedure were used in the CNC-based production of three types of nine beams: three reference rectangular beams, three Vmin beams, and three Qmax beams. All specimens were subjected to a four-point bending test. The experimental results were contrasted with theoretical assumptions. Optimization reduced material usage by ca. 12.9% while preserving approximately the same LCC. The maximization of LCC was found to be rather unsuccessful due to the significant dependence of the beams' response on the highly variable mechanical properties of GLT.

Optimal strategies for investment and consumption: stochastic analysis with pontryagin's principle under economic uncertainty

This study investigates how stochastic optimization is applied to the management of a company's portfolio in order to maximize the expected utility of wealth over a given period. Inspired by Merton's research, this model involves random volatility in the financial markets, while maintaining a constant interest rate to take better account of real economic uncertainties. The aim is to formulate optimal investment and consumption strategies based on Pontryagin's maximum principle. Taking into account key factors such as economic growth and market volatility, as well as risk aversion in our financial considerations, we recommend an approach incorporating a quadratic penalty for excessive investment. This innovative method aims to adjust financial choices in line with economic fluctuations and ensure prudent management of the company's monetary resources. Finally, numerical simulations illustrate the influence of these factors on overall wealth, as well as on investment and consumption, underlining the importance of prudent portfolio management during periods of uncertainty.

Optimal control and cost-effectiveness analysis for bacterial meningitis disease

In this study, we developed an optimal control deterministic model for the dynamics of bacterial meningitis disease. The objective was to investigate the efficiency and cost-effectiveness of the three controls (prevention, treatment, and screening) in curtailing the spread of bacterial meningitis. To accomplish this, we applied Pontryagin's maximum principle to derive the optimality system. We examined different combination strategies to investigate the effect of the interventions on the spread of bacterial meningitis. We used an incremental cost-effectiveness ratio (ICER) to examine which control technique was the most successful. The simulation results show that combining prevention and screening is the most cost-effective strategy. The objective function and the incremental cost-effectiveness ratio further support this result, indicating that maximum utilization of prevention and screening is required for the entire period.

MATHEMATICAL MODELING AND OPTIMAL CONTROL OF WHEAT STRIPE RUST (Yellow Rust) DISEASE

Wheat Stripe Rust is a devastating fungal disease that is the largest limitation to wheat production and threatens the global food supply. In this work, we proposed a mathematical system of ordinary differential equations to model the wheat Stripe Rust transmission dynamics. First, we showed the feasible region, the positivity of the solution, the model’s equilibrium points and the basic reproduction number. Second, we used the Pontryagin maximum principle to extend the basic system into an optimal control system by embedding three control measures (resistant cultivars, fungicides and cultural practices). Finally, we used a numerical simulation of the optimality system to demonstrate the thorough impacts of resistant wheat cultivars, fungicide treatments and cultural practices in reducing the epidemic.

Optimal control analysis applied to the transmission dynamics of Hepatitis B to primary liver cancer

Infection with hepatitis virus, especially hepatitis B virus causes an irritation to the liver. Ranked among the top ten diseases with high mortality rate, viral hepatitis poses a great health challenges worldwide, with threat to chronic infection, hepatitis-related liver cancer and cirrhosis. Hence, a mathematical model to study the dynamics of hepatitis B virus infection from progressing into primary liver cancer was developed for analysis. We aimed at obtaining the optimal control strategies needed to reduce the number of new cases of this disease, and also reducing the deterioration rate of people living with this disease from sliding into primary liver cancer. We introduced four distinct control variables at each point in the model, and assumed that all the controls are set of Lebesque Measurable functions. The Pontryagin’s maximum principle (PMP) is employed to establish the optimal effect of these controls on the disease under study. Existence of model solution was established using the appropriate theorem. The model with control strategies was analytically solved using PMP and numerically simulated for each compartments, to establish the effect of the control variables on the dynamics of transmission of this infection within the compartment, and their overall effect on the entire population. Differential transform method (DTM) was later adopted as a semi-analytic scheme to solve the developed model. The series solution of DTM was numerically plotted for each compartment and compared with Runge-Kutta order 4 (RK4) numerical scheme. Analysis of the model with control pinpoint the importance of sensitization and vaccination on the overall dynamics of the infection, while the numerical plot of DTM and RK4 established the efficacy of the adopted semi-analytic method (DTM) to accurately solve the system of equations of the model.

Optimizing control strategies for monkeypox through mathematical modeling

Monkeypox is a zoonotic viral disease similar to smallpox, has emerged as a major global health concern following the COVID-19 pandemic. This study presents a novel mathematical model aimed at analyzing various epidemiological factors, particularly the less-explored transmission from humans to monkeys, where both species act as carriers. Our approach integrates comprehensive awareness campaigns, strict security measures, and targeted health interventions to limit transmission between hosts, with the goal of reducing human infections and eliminating the virus among animal populations. The model utilizes the continuous-time Pontryagin maximum principle to determine and apply optimal control strategies, with iterative simulations conducted in Matlab. Our results, derived from these simulations, show that implementing all proposed preventative strategies—such as public awareness efforts, isolation of infected monkeys, and vaccination—simultaneously is the most effective method to control the virus’s spread. We observed a significant reduction in both human and animal infections when these strategies were combined. The study’s conclusions provide important insights into the transmission dynamics of monkeypox, highlighting the critical role of multifaceted intervention strategies in controlling outbreaks. These findings are expected to support more effective public health management and contribute to the global effort to contain and ultimately eradicate monkeypox.

Low Computational Cost for Multiple Waypoints Trajectory Planning: A Time‐Optimal‐Based Approach

In the field of mobile robots, achieving minimum time in executing trajectories is crucial for applications like delivery, inspection, and search and rescue. In this article, a novel time‐optimal planner based on optimization methods is introduced. Despite the high computational cost associated with such methods, the solution calculates time‐optimal multi‐waypoint trajectories, achieving results in the order of milliseconds. The proposed method formulates a time‐optimal trajectory using the Pontryagin's maximum principle as a policy. By utilizing a point mass model, the planner generates trajectories that are adaptable to different robot models. The approach incorporates a definition of a search space to guarantee convergence while considering the system limits. Simulation and real‐world experiments are performed to validate the feasibility of our method with different configurations. Simulation results compared to a benchmark method demonstrate our approach's superior performance in terms of computational time, achieving near‐optimal solutions. In addition, in the real‐world experiments, the integration of the method into practical applications is validated.

Machine learning-based optimal temperature management model for safety and quality control of perishable food supply chain

The management of a food supply chain is difficult and complex because of the product's short shelf-life, time-sensitivity, and perishable nature which must be carefully considered to minimize food waste. Temperature-controlled perishable food supply chain provides the highly crucial facilities necessary to maintain the quality and safety of the product. The storage temperature is the most vital factor in maintaining both the quality and shelf-life of a perishable food. Adequate storage temperature control ensures that perishable foods are transported to the end-users in good quality and safe to consume. This paper presents perishable food storage temperature control through mathematical optimal control model where the storage temperature is regarded as the control variable and the deterioration of the perishable food’s quality follows the first-order reaction. The optimal storage temperature for a single perishable food is determined by applying the Pontryagin's maximum principle to solve the optimal control model problem. For multi-temperature commodities supply chain, an unsupervised machine learning (ML) method, called k-means clustering technique is used to determine the temperature clusters for a range of perishables. Based on descriptive analysis, it is observed that the k-means clustering technique is effective in identifying the best suitable storage temperature clusters for quality control of multi-commodity supply chain.

Dynamic pricing and inventory model for perishable products with reference price and reference quality

Reference price (quality) is a benchmark point used by consumers to make price (quality) judgements. Combining reference price and reference quality with dynamic pricing and inventory management, this paper applies differential equations theory to construct a optimal control model for perishable products when the quality of products declines exponentially. The demand of products depends on price, quality, reference price and reference quality, among which reference price and reference quality are influenced by consumers’ past memory. The aim is to maximize the retailer’s profit during the period. The conclusions are as follows. Firstly, a optimal dynamic pricing and inventory model for perishable products with reference price and reference quality is constructed, and the model is extended to dual decision variables, stochastic demand, time-dependent effects, competitive and infinite planing horizon setups. Secondly, through the Pontryagin maximum principle, the analytical expression of the optimal dynamic price is derived. Thirdly, a linear search method to solve the optimal static price is proposed. Finally, the sensitivity of the main parameters is analyzed and the corresponding management enlightenments are given. By comparison of dynamic and static pricing, we find that dynamic pricing can achieve more profits and take a shorter selling period. In addition, for the sales problem of perishable products affected by both reference price and reference quality, retailers should adopt skimming pricing strategy (the optimal price decreases with time). Furthermore, to obtain more profit, retails should strive to increase the sensitivity coefficients of two types of reference, and the reference quality memory coefficient, while to decrease the reference price memory coefficient.

Nonlinear Optimal Control Based on FBDEs and its Application to AGV.

This article focuses on solving a finite-horizon nonlinear optimal control problem by using the Pontryagin's maximum principle. In practical applications, linearization is a common approach for solving nonlinear dynamical systems. However, it is not universally applicable due to various reasons, such as instability and low accuracy. In contrast to linearization, the inherent challenge in directly solving the above nonlinear optimal control problem lies in addressing the highly coupled nonlinear forward and backward differential equations. In order to address this problem, an equivalent relationship is established between these equations and a new optimization problem. By exploiting the inherent relationship between supervised learning and an optimization problem from the view of a dynamical system, a deep neural network framework is constructed for describing the new optimization problem. Furthermore, a numerical algorithm for optimal control, which is very powerful for a large variety of nonlinear dynamical systems, is implemented by training a deep residual network. Finally, the effectiveness of the algorithm is demonstrated by solving a trajectory tracking control problem for automatic guided vehicle. The obtained results reveal that the proposed control scheme can achieve high-precision tracking.

Optimal Control Analysis of Isolation and Treatment Strategies on the Transmission Dynamics and Control of Lassa Fever

This study constructed and examined a mathematical model on the dynamics of Lassa fever transmission and control. We took into account two interacting rodent and human populations. There are six groups within the human population, one of which is made up of people who underwent complicated recovery. Additionally, there are three divisions within the rodent population. The reduction of susceptible, latent, and infectious individuals by treatment and isolation is the main goal of this work. The goal of optimality is to reduce the number of latent and infectious individuals through isolation and treatment. The amount of influence of the controls in lowering the population of causal pathogens and susceptible, latent, and infectious people is determined by applying the optimal control theory. The model's optimal solution was established using Pontryagin's maximum principle, and the optimal system was then generated and solved numerically. Graphs were used to create simulations showing how the controls affected susceptible, latent, and infected persons. The results demonstrated that applying the two controls simultaneously can be one of the quick and efficient strategies to regulate Lassa Fever.

Time Parametrized Motion Planning

Time can be treated as a free parameter to isotropically stretch the tangent space. A trajectory, which matches the boundary conditions on its configuration, is adjusted so that velocity conditions are met. The modified trajectory is found by substitution, without the computational cost of re-integrating the velocity function. This concept is extended to stretch the tangent space anisotropically. This method of time parametrization especially applies to Geometric Control, where the Pontryagin Maximum Principle minimizes some cost function and matches the boundary configuration constraints but not the velocity constraints. The optimal trajectory is modified by the parametrization so that the cost function is minimized if the stretching is stopped at any time. This is a theoretical contribution, using a wheeled robot example to illustrate the modification of an optimal velocity under multiple parametrizations.

A dynamical optimal control theory and cost-effectiveness analyses of the HBV and HIV/AIDS co-infection model.

Studies have shown that the co-infection of Human Immunodeficiency Virus (HIV) and Hepatitis B Virus (HBV) poses a major threat to the public health due to their combined negative impacts on health and increased risk of complications. Even though, some scholars formulated and analyzed the HBV and HIV co-infection model they did not consider the compartment that contains protected individuals against both HBV and HIV infections. They incorporated the optimal control theory and cost-effectiveness analysis simultaneously. With this in mind, we are motivated to formulate and analyze the HBV and HIV co-infection model, considering the protected group and incorporating optimal control theory and cost-effectiveness. In this study, we have theoretically computed all of the models disease-free equilibrium points, all the models effective reproduction numbers and unique endemic equilibrium points. The two sub-models disease-free equilibrium points are locally as well as globally asymptotically stable whenever their associated effective reproduction numbers are less than one. We reformulated the optimal control problem by incorporating five time-dependent control measures and conducted its theoretical analysis by utilizing the Pontryagin's maximum principle. Using the fourth order Runge-Kutta numerical method and MATLAB ODE45, we performed the numerical simulations with various combinations of control efforts to verify the theoretical results and investigate the impacts of the suggested protection and treatment control strategies for both the HBV and HIV diseases. Also, we carried out a cost-effectiveness analysis of the proposed control strategies. Eventually, we compared our model results with other researcher similar model results whenever cost-effectiveness analysis is not carried out the findings of this particular study suggest that implementing each of the proposed control strategies simultaneously has a high potential to reduce and control the spread of HBV and HIV co-infections in the community. According to the cost-effectiveness analysis, implementing the HBV treatment and the HIV and HBV co-infection treatment measures has a high potential effect on reducing and controlling the HBV and HIV co-infection transmission problem in the community.

Investigation of an optimal control strategy for a cholera disease transmission model with programs

Cholera is a disease of poverty affecting people with inadequate access to safe water and basic sanitation. Conflict, unplanned urbanization and climate change all increase the risk of cholera. In this article, an optimal control deterministic mathematical model of cholera disease with cost-effectiveness analysis is developed and analyzed considering both direct and indirect contact transmission pathways. The model qualitative behaviors, such as the invariant region, the existence of a positive invariant solution, the two equilibrium points (disease-free and endemic equilibrium), and their stabilities (local as well as global stability) of the model are studied. Moreover, the basic reproduction number of the model is obtained. We also performed sensitivity analysis of the basic parameters of the model. Then an optimal control problem is designed with a control functional having five controls: vaccination, treatment, environment sanitation and personal hygiene, and water quality improvement program. We examined the existence and uniqueness of the optimal controls of the system. Through the implementation of Pontryagin's maximum principle, the characterization of the optimal controls optimality system is established. The numerical simulation results the integrated control strategies demonstrated that strategy 2, 7, and 12 are effective programs to combat cholera disease from the community. Based on the local circumstances, available funds, and resources, it is recommended to the government stakeholders and policymakers to execute any one of the three integrated intervention programs.

Optimal quantum controls robust against detuning error

Abstract Precise control of quantum systems is one of the most important milestones for achieving practical quantum technologies, such as computation, sensing, and communication. Several factors deteriorate the control precision and thus their suppression is strongly demanded. One of the dominant factors is systematic errors, which are caused by discord between an expected parameter in control and its actual value. Error-robust control sequences, known as composite pulses, have been invented in the field of nuclear magnetic resonance (NMR). These sequences mainly focus on the suppression of errors in one-qubit control. The one-qubit control, which is the most fundamental in a wide range of quantum technologies, often suffers from detuning error. As there are many possible control sequences robust against the detuning error, it will practically be important to find ‘optimal’ robust controls with respect to several cost functions such as time required for operation, and pulse-area during the operation, which corresponds to the energy necessary for control. In this paper, we utilize the Pontryagin’s maximum principle (PMP), a tool for solving optimization problems under inequality constraints, to solve the time and pulse-area optimization problems. We analytically obtain pulse-area optimal controls robust against the detuning error. Moreover, we found that short-CORPSE, which is the shortest known composite pulse so far, is a probable candidate of the time optimal solution according to the PMP. We evaluate the performance of the pulse-area optimal robust control and the short-CORPSE, comparing with that of the direct operation.

Ecological‐economic modeling of nutrient enrichment in multispecies fisheries

AbstractIn fisheries economics, multispecies or ecosystem models typically focus on interactions among fish species, often overlooking the flow of nutrients. However, biologists widely acknowledge that nutrients–Âİsuch as nitrogen, phosphorus, potassium, and calcium–Âİare vital for organisms and play crucial roles in ecosystem dynamics. This paper delves into a bioeconomic predator‐prey model incorporating nutrient enrichment through Holling Type II and Michaelis–Menten functional forms. Initially, we examine the system's dynamics before formulating an optimal time‐dependent harvest policy using the Pontryagin maximum principle. Our findings reveal that heightened nutrient enrichment can lead to the breakdown of a positive stable equilibrium, resulting in the extinction of both prey and predator. The stability of the ecosystem may also be influenced by the interplay of nutrient enrichment and predator harvest.