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This article is concerned with optimal control problems for plants described by systems of high order nonlinear PDE's (whose linear approximation is elliptic in the sense of Douglis-Nirenberg), with a special attention being given to their particular case: the standard stationary system of non-linear Navier–Stokes equations. The objective is to establish an analog of the Pontryagin's maximum principle. The major challenge stems from the presence of infinitely many point-wise constraints on the system's state, which are imposed at any point from a given continuum set of independent variables. Necessary conditions for optimality in the form of an "abstract" maximum principle are first obtained for a general optimal control problem couched in the language of functional analysis. This result is targeted at a wide class of problems, with an idea to absorb, in its proof, a great deal of technical work needed for derivation of optimality conditions so that only an interpretation of the discussed result would be basically needed to handle a particular problem. The applicability of this approach is demonstrated via obtaining the afore-mentioned analog of the Pontryagin's maximum principle for a state-constrained system of high-order elliptic equations and the Navier–Stokes equations. [ABSTRACT FROM AUTHOR]

In this article, a finite‐horizon adaptive Pontryagin's maximum principle is presented for nonlinear systems with state inequality constraints. Concurrent learning (CL) technique is adopted to identify the unknown parameters of the dynamic systems. Based on the identification model, a novel adaptive iterative algorithm under the Pontryagin's framework is introduced to learn the finite‐horizon optimal control solution. Convergence analysis of the algorithm is provided by showing that the cost function sequence is monotonically decreasing. Furthermore, we extend the adaptive iterative algorithm to time‐varying nonlinear systems. The new algorithm overcomes the technical obstacles of the existing adaptive/approximate dynamic programming (ADP) approaches to deal with the time‐varying characteristic of Hamilton–Jacobi–Bellman (HJB) partial differential equation (PDE), especially when state constraints exist. Simulation examples are carried out to validate the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

In pursuit–evasion scenarios, the pursuer typically possesses a lethal zone. If the evader effectively utilizes perceptual information, they can narrowly escape the lethal zone while minimizing energy consumption, thereby avoiding excessive and unnecessary maneuvers. Based on optimal control theory, we propose a guidance law for achieving critical safe miss distance evasion under bounded control. First, we establish the zero-effort miss (ZEM) state equation for the evader, while approximating disturbances from the pursuer. Next, we formulate an optimal control problem with energy consumption as the objective function and the ZEM at the terminal time as the terminal constraint. Subsequently, we design an iterative algorithm that combines the homotopy method and Newton's iteration to solve the optimal control problem, applying Pontryagin's Maximum Principle. The simulation results indicate that the designed iterative method converges effectively; through online updates, the proposed guidance law can successfully achieve critical safe miss distance evasion. Compared to programmatic maneuvering and norm differential game guidance law, this approach not only stabilizes the evader's evasion capabilities but also significantly reduces energy consumption. [ABSTRACT FROM AUTHOR]

This article explores two distinct issues. To begin with, we analyze the Pontriagin maximum principle concerning fractional delay differential equations. Furthermore, we investigate the most effective method for resolving the control problem associated with Eq. (1.1) and its corresponding payoff function (1.2). Subsequently, we explore the Pontryagin Maximum principle within the framework of Volterra delay integral equations (1.3). We enhance the outcomes of our investigation by presenting illustrative examples towards the conclusion of the article. [ABSTRACT FROM AUTHOR]

Abstract Faba bean is one of the most important grown plants worldwide for human and animal. Despite its various importance, the productivity of faba bean has been constrained by several biotic and abiotic factors. Many faba bean pathogens have been reported so far, of which the most important yield limiting disease is Chocolate Spot Disease (Botrytis fabae). The dynamics of disease transmission and decision-making processes for intervention programs for disease control are now better understood through the use of mathematical modeling. In this paper a deterministic mathematical model for Chocolate Spot disease (CSD) on faba bean plant with an optimal control model was developed and analyzed to examine the best strategy in controlling CSD. The optimal control model is developed with three control interventions, namely prevention ( $$u_{1}$$ u 1 ), quarantine ( $$u_{2}$$ u 2 ) and chemical control ( $$u_{3}$$ u 3 ). The Pontryagin’€™s maximum principle isused to derive the Hamiltonian, the adjoint variables, the characterization of the controls and the optimality system. A cost-effective approach is chosen from a set of possible integrated strategies using the incremental cost-effectiveness ratio (ICER). The forward-backward sweep iterative approach is used to run numerical simulations. We obtained the Hamiltonian, the adjoint variables, the characterization of the controls and the optimality system. The numerical results demonstrate that each integrated strategy can reduce the diseases within the specified period. However due to limited resources, an integrated strategy prevention and uprooting was found to be a best cost-effective strategy to combat CSD. Therefore, attention should be given for the integrated cost-effective and environmentally eco-friendly strategy by stake holders and policy makers to control CSD and disseminate the integrated intervention to the farmers in order to fight the spread of CSD in the Faba bean population and produce the expected yield from the field.

The optimal control problem for a nonlinear dynamic system of a cascade type with endpoint and irregular pointwise state constraints (the so-called state constraints of depth 2) is studied. This problem admits a refined formulation of Pontryagin’s maximum principle in terms of a (non-standard) Hamilton-Pontryagin function of the second order. The question of estimating the jump of the derivative of the Lagrange multiplier corresponding to the state constraint is studied. Some sufficient conditions have been obtained under which the maximum principle implies uniform in time estimates for the jump of the specified function. In particular, sufficient conditions have been given for the absence of a jump (i.e., continuous differentiability) of the multiplier. These results are based on the concepts of 2-regularity of the state constraint and the so-called regularity zone of the problem. The obtained estimates are of interest for the theory of Pontryagin’s maximum principle and can be used in practice, including the implementation of the known shooting method within the framework of one of the standard approaches to the numerical interpretation of the necessary optimality condition.

Faba bean is one of the most important grown plants worldwide for human and animal. Despite its various importance, the productivity of faba bean has been constrained by several biotic and abiotic factors. Many faba bean pathogens have been reported so far, of which the most important yield limiting disease is Chocolate Spot Disease (Botrytis fabae). The dynamics of disease transmission and decision-making processes for intervention programs for disease control are now better understood through the use of mathematical modeling. In this paper a deterministic mathematical model for Chocolate Spot disease (CSD) on faba bean plant with an optimal control model was developed and analyzed to examine the best strategy in controlling CSD. The optimal control model is developed with three control interventions, namely prevention ( u 1 ), quarantine ( u 2 ) and chemical control ( u 3 ). The Pontryagin'€™s maximum principle isused to derive the Hamiltonian, the adjoint variables, the characterization of the controls and the optimality system. A cost-effective approach is chosen from a set of possible integrated strategies using the incremental cost-effectiveness ratio (ICER). The forward-backward sweep iterative approach is used to run numerical simulations. We obtained the Hamiltonian, the adjoint variables, the characterization of the controls and the optimality system. The numerical results demonstrate that each integrated strategy can reduce the diseases within the specified period. However due to limited resources, an integrated strategy prevention and uprooting was found to be a best cost-effective strategy to combat CSD. Therefore, attention should be given for the integrated cost-effective and environmentally eco-friendly strategy by stake holders and policy makers to control CSD and disseminate the integrated intervention to the farmers in order to fight the spread of CSD in the Faba bean population and produce the expected yield from the field. [ABSTRACT FROM AUTHOR]

This contribution shows a geometric optimal control procedure to solve the trajectory generation problem for the navigation (generic motion) of mechanical systems with revolute joints. The mechanical system is analyzed as a nonlinear Lagrangian system affected by dry friction at the joint level. Rayleigh's dissipation function is used to model this dissipative effect of joint-level friction, and regarded as a potential. Rayleigh's potential is an invariant scalar quantity from which friction forces derive and are represented by a smooth model that approaches the traditional Coulomb's law in our proposal. For the optimal control procedure, an invariant cost function is formed with the motion equations and a Riemannian metric. The goal is to minimize the consumed energy per unit time of the system. Covariant control equations are obtained by applying Pontryagin's principle, and time-integrated using a Finite Elements Method-based solver. The obtained solution is an optimal trajectory that is then applied to a mechanical system using a proportional–derivative plus feedforward controller to guarantee the trajectory tracking control problem. Simulations and experiments confirm that including joint-level friction forces at the modeling stage of the optimal control procedure increases performance, compared with scenarios where the friction is not taken into account, or when friction compensation is performed at the feedback level during motion control. [ABSTRACT FROM AUTHOR]

This study examines the scientific production focused on the Maximum Principle between 1962 and 2021. Results indicate a consistent increase in the absolute number of publications over time. In relative terms, there is a resurgence of interest in this research field after the period between 2004 and 2009. Overall, these findings support the idea of strategic complementarity between the Maximum Principle and optimal control. However, there is a notable exception during the period 2010–2015, characterised by a decline in scientific production focused on the Maximum Principle and a simultaneous increase in focus on optimal control. Academic journals that play a role in promoting this research field tend to have high impact factors and interesting cite scores. Using a modified Boston Consulting Group matrix, the results reveal the persistence of two researchers labelled as stars and three as cash cows. A multiple linear regression analysis confirms that reputation significantly influences the clustering trends. A critical discussion is provided to highlight the dichotomy between popularity and effective contributions in this research field. [ABSTRACT FROM AUTHOR]

In this paper, we conducted a study on the optimal control problem of an epidemic model which consists of two strain with different types of incidence rates: bilinear and non-monotonic. We also considered use of the saturation treatment function. Two basic regeneration numbers are calculated from the epidemic model, which are denoted as R 1 and R 2 . The global stability of the disease-free equilibrium point was studied by the Lyapunov method, and it was proved that the disease-free equilibrium point is globally asymptotically stable when R 1 and R 2 are less than one. Finally, we formulated a time-dependent optimal control problem by Pontryagin's maximum principle. Numerical simulations were performed to establish the effects of model parameters for disease transmission as well as the effects of control. [ABSTRACT FROM AUTHOR]

This work studies a class of singular Volterra integral equations that are (controlled) and can be applied to memory-related problems. For optimum controls, we prove a second-order Pontryagin type maximal principle. [ABSTRACT FROM AUTHOR]

Improving epidemic models to better reflect reality has long been a prominent concern for governments and researchers. This paper presents a novel Susceptible–Infected–Recovered–Susceptible (SIRS) epidemic model for human populations, offering a comprehensive analysis. The proposed model introduces a generalized SIRS epidemics framework encompassing three propagation scenarios. The paper establishes the positivity and boundedness of the system and demonstrates the stability of its equilibrium points. Furthermore, a controlled system is introduced, accompanied by three suggested control strategies to minimize the infected population while optimizing cost. To validate the analytical findings, a numerical example is provided. The paper concludes with a summary and outlines future research directions. [ABSTRACT FROM AUTHOR]

In this paper we advocate for Isaacs' method for the solution of differential games to be applied to the solution of optimal control problems. To make the argument, the vehicle employed is Pontryagin's canonical optimal control example, which entails a double integrator plant. However, rather than controlling the state to the origin, we require the end state to reach a terminal set that contains the origin in its interior. Indeed, in practice, it is required to control to a prescribed tolerance rather than reach a desired end state; constraining the end state to a terminal manifold of co-dimension n − 1 renders the optimal control problem easier to solve. The global solution of the optimal control problem is obtained and the synthesized optimal control law is in state feedback form. In this respect, two target sets are considered: a smooth circular target and a square target with corners. Closed-loop state-feedback control laws are synthesized that drive the double integrator plant from an arbitrary initial state to the target set in minimum time. This is accomplished using Isaacs' method for the solution of differential games, which entails dynamic programming (DP), working backward from the usable part of the target set, as opposed to obtaining the optimal trajectories using the necessary conditions for optimality provided by Pontryagin's Maximum Principle (PMP). In this paper, the case is made for Isaacs' method for the solution of differential games to be applied to the solution of optimal control problems by way of the juxtaposition of the PMP and DP methods. [ABSTRACT FROM AUTHOR]

In order to maintain the battery SOC, the fuel cell power will fluctuate dramatically, as well as frequent start-stop, which will greatly increase the life attenuation of the fuel cell and reduce the durability. An optimization-based energy management strategy with a real-time model predictive control and pontryagin's maximum principle for FCHEV is proposed in this paper, both the fuel economy and the fuel cell durability are considered in the optimization. A novel model predictive control is studied to achieve energy distribution. After the calculation of predicted speed sequence through back propagation neural network, pontryagin's maximum principle is introduced to solve the optimal control problem in each prediction horizon and obtain the ideal control strategy. In addition, the fuel cell degradation model is introduced in the modeling process, the minimum power point of the fuel cell system is designed to improve the fuel economy and durability of the fuel cell. Compared with the rule-based strategy, the proposed MPC strategy has better performance to reduce the total equivalent hydrogen consumption, which can save up to 8.44% in the test case of the mid-size fuel cell passenger car while maintaining the stability of the battery's state of charge. [ABSTRACT FROM AUTHOR]

We study the problem of optimal stimulation of demand based on a controlled version of Kaldor's business cycle model. Using the approximation method, we prove a version of Pontryagin's maximum principle in normal form containing an additional pointwise condition on the adjoint variable. The results obtained develop and strengthen the previous results in this direction. [ABSTRACT FROM AUTHOR]

We consider a finite number of N statistically equal agents, each moving on a finite set of states according to a continuous-time Markov Decision Process (MDP). Transition intensities of the agents and generated rewards depend not only on the state and action of the agent itself, but also on the states of the other agents as well as the chosen action. Interactions like this are typical for a wide range of models in e.g. biology, epidemics, finance, social science and queueing systems among others. The aim is to maximize the expected discounted reward of the system, i.e. the agents have to cooperate as a team. Computationally this is a difficult task when N is large. Thus, we consider the limit for N → ∞. In contrast to other papers we treat this problem from an MDP perspective. This has the advantage that we need less regularity assumptions in order to construct asymptotically optimal strategies than using viscosity solutions of HJB equations. The convergence rate is 1 / N . We show how to apply our results using two examples: a machine replacement problem and a problem from epidemics. We also show that optimal feedback policies from the limiting problem are not necessarily asymptotically optimal. [ABSTRACT FROM AUTHOR]

This work studies a class of singular Volterra integral equations that are (controlled) and can be applied to memory-related problems. For optimum controls, we prove a second-order Pontryagin type maximal principle.

Extremists are increasingly using social media to recruit and radicalize other users and increase their money. Terrorists can use popular social networks accounts and perform their activities in a hidden way. So, it is crucial to create a fruitful mechanism for controlling the spread of misinformation. Otherwise, a large number of people can mislead by this terrorist activity by joining them. Here, we propose malicious news spreading model incorporating hidden attackers of a social network. A threshold is defined for deciding the extinction of malicious news from a social network. Here, we show the importance of network alertness and activity of cybersecurity agencies in the modified model. Moreover, we obtained the optimal values of the control parameters for emergencies.

In this paper, we study the modern mathematical theory of the optimal control problem associated with the fractional Roesser model and described by Caputo partial derivatives, where the functional is given by the Riemann-Liouville fractional integral. In the formulated problem, a new version of the increment method is applied, which uses the concept of an adjoint integral equation. Using the Banach fixed point principle, we prove the existence and uniqueness of a solution to the adjoint problem. Then the necessary and sufficient optimality condition is derived in the form of the Pontryagin’s maximum principle. Finally, the result obtained is illustrated by a concrete example.

Abstract In the last two decades, Nipah virus (NiV) has emerged as a significant paramyxovirus transmitted by bats, causing severe respiratory illness and encephalitis in humans. NiV has been included in the World Health Organization’s Blueprint list of priority pathogens due its potential for human-to-human transmission and zoonotic characteristics. In this paper, a mathematical model is formulated to analyze the dynamics and optimal control of NiV. In formulation of the model we consider two modes of transmission: human-to-human and food-borne. Further, the impact of contact with an infected corpse as a potential route for virus transmission is also consider in the model. The analysis identifies the model with constant controls has three equilibrium states: the NiV-free equilibrium, the infected flying foxes-free equilibrium, and the NiV-endemic equilibrium state. Furthermore, a theoretical analysis is conducted to presents the stability of the model equilibria. The model fitting to the reported cases in Bangladesh from 2001 to 2015, and the estimation of parameters are performed using the standard least squares technique. Sensitivity analysis of the model-embedded parameters is provided to set the optimal time-dependent controls for the disease eradication. The necessary optimality conditions are derived using Pontryagin’s maximum principle. Finally, numerical simulation is performed to determine the most effective strategy for disease eradication and to confirm the theoretical results.

Abstract Fuel consumption is one of the main concerns for heavy-duty trucks. Predictive cruise control (PCC) provides an intriguing opportunity to reduce fuel consumption by using the upcoming road information. In this study, a real-time implementable PCC, which simultaneously optimizes engine torque and gear shifting, is proposed for heavy-duty trucks. To minimize fuel consumption, the problem of the PCC is formulated as a nonlinear model predictive control (MPC), in which the upcoming road elevation information is used. Finding the solution of the nonlinear MPC is time consuming; thus, a real-time implementable solver is developed based on Pontryagin’s maximum principle and indirect shooting method. Dynamic programming (DP) algorithm, as a global optimization algorithm, is used as a performance benchmark for the proposed solver. Simulation, hardware-in-the-loop and real-truck experiments are conducted to verify the performance of the proposed controller. The results demonstrate that the MPC-based solution performs nearly as well as the DP-based solution, with less than 1% deviation for testing roads. Moreover, the proposed co-optimization controller is implementable in a real-truck, and the proposed MPC-based PCC algorithm achieves a fuel-saving rate of 7.9% without compromising the truck’s travel time.

Abstract The leaves of many trees emit volatile organic compounds (abbreviated as BVOCs), which protect them from various damages, such as herbivory, pathogens, and heat stress. For example, isoprene is highly volatile and is known to enhance the resistance to heat stress. In this study, we analyze the optimal seasonal schedule for producing isoprene in leaves to mitigate damage. We assume that photosynthetic rate, heat stress, and the stress-suppressing effect of isoprene may vary throughout the season. We seek the seasonal schedule of isoprene production that maximizes the total net photosynthesis using Pontryagin’s maximum principle. The isoprene production rate is determined by the changing balance between the cost and benefit of enhanced leaf protection over time. If heat stress peaks in midsummer, isoprene production can reach its highest levels during the summer. However, if a large portion of leaves is lost due to heat stress in a short period, the optimal schedule involves peaking isoprene production after the peak of heat stress. Both high photosynthetic rate and high isoprene volatility in midsummer make the peak of isoprene production in spring. These results can be clearly understood by distinguishing immediate impacts and the impacts of future expectations.

In epidemic prevention efforts, the emergence of new virus strains due to mutations greatly complicates the prediction and management of epidemics. Most of the current mathematical models of infectious diseases assume that the mutant strain and the original strain have the same outbreak time, which is obviously an ideal situation. In order to make the study more practical, we consider the general situation of outbreaks of mutated strains. At the same time, the optimal control strategy under different emergence time of mutant strains was proposed by using the optimal control theory and numerical simulation. This study provides a new theoretical framework for the dual strain competition model with different outbreak times. The final theoretical results and numerical simulation showed that although the emergence time of the mutant did not affect the final trend of the epidemic, it would affect the cost of prevention and control during the control period. [ABSTRACT FROM AUTHOR]

The fashion apparel industry is facing an increasingly growing demand, compounded by the short sales lifecycle and strong seasonality of clothing, posing significant challenges to inventory management in the retail sector. Despite some retailers like Uniqlo and Zara implementing inventory management and dynamic pricing strategies, challenges persist due to the dynamic nature of fashion trends and the stochastic factors affecting inventory. To address these issues, we construct a mathematical model based on the mathematical expression of the deterministic fashion level function, where the geometric Brownian motion, widely applied in finance, is initially utilized in the stochastic fashion level function. Drawing on research findings from deteriorating inventory management and stochastic optimization, we investigate the fluctuation of inventory levels, optimal dynamic pricing, optimal production rates, and profits—four crucial indicators—via Pontryagin's maximum principle. Analytical solutions are derived, and the numerical simulation is provided to verify and compare the proposed model with deterministic fashion level function models. The model emphasizes the importance of considering stochastic factors in decision-making processes and provides insights to enhance profitability, inventory management, and sustainable consumption in the fashion product industry. [ABSTRACT FROM AUTHOR]

In the present work, we developed a deterministic SHATR (Susceptible - HIV infected -AIDS infected - Antiretroviral Treatment - Recovered) compartment model for HIV/AIDS. This model considers the disease outbreak due to a lack of awareness and treatment. The steady states of the proposed model system are obtained and analyzed by using the nonlinear stability theory of differential equations. The basic reproduction number is derived and explored to determine the stability and sensitivity index of some important relative parameters. Further, to know the global behavior of the model one parameter bifurcation study is discussed. Moreover, the optimal control theory has been applied to identify the optimal strategy by taking treatment and awareness for safe intercourse as control parameters. The control problem is solved analytically by using Pontryagin's maximum principle. Finally, the model is simulated to describe the optimality under various assumptions and the stability of equilibrium points. [ABSTRACT FROM AUTHOR]

The time-optimal control problem for an arbitrary number of nonsynchronous oscillators with a limited scalar control is considered. An analytical investigation of the problem is performed. The property of strong accessibility and global controllability is proved, and a program control is found that brings the system from the origin to a fixed point in the shortest time. Trajectories satisfying both the motion equations of the system and the additional conditions based on the matrix nondegeneracy conditions of the relay control have been found for bringing a group of oscillators to the origin. Two classification methods of trajectories according to the number of control switchings are compared: the one based on the necessary extremum conditions and the Neustadt–Eaton numerical algorithm. [ABSTRACT FROM AUTHOR]

Nowadays, environmental issues have received increasing attention from experts. The main cause is the increase of carbon emissions in the atmosphere, so it is urgent to reduce carbon emissions. In order to establish the optimal pricing strategy as well as the emission reduction effort strategy for companies who produce and sell low carbon products, this paper proposes an optimal control model for low carbon products. The reduction of the carbon emission for the product is described dynamically by a differential equation, and the analytical expressions of the optimal pricing and the emission abatement strategies are derived using the Pontryagin's maximum principle. Finally, the numerical experiments are used to explain the results obtained. The results show that companies producing and selling low-carbon products must pay more attention to the amount of carbon emission reduction in their products, and make more efforts to reduce emissions in order to make more profits. Additionally, the parametric analysis shows that expanding market size and reducing inventory depletion can be equally helpful in shortening the sales cycle and boosting profits. [ABSTRACT FROM AUTHOR]

A proposed two-dimensional modified Lotka-Volterra fishery model in terms of predator-prey aims to explore the effect of non-selective harvesting on the predator and prey populations. The study delves into various essential aspects of the dynamical system, including positivity, uniform boundedness, and persistence. Points of equilibrium are identified. The system's local and global stability are thoroughly examined and discussed. Moreover, the research explores the concept of bionomic equilibrium, a scenario where economic rent is entirely dissipated. Extending the bioeconomic model, the study investigates a linear optimal control problem to determine the most effective harvesting strategy. Utilising Pontryagin's maximum principle, the optimal control is characterised. The findings indicate that maximum allowable effort should be exerted whenever the net revenue per unit effort surpasses the total net marginal revenue of predator and prey stocks. Numerical simulations, using data on the marine artisanal fishery in Ghana, are conducted to validate the theoretical discoveries. The outcomes reveal that the fishery can support sustainable harvesting of both predator (tuna) and prey (sardinella) populations, as long as the optimal harvesting effort is set at 100,000 fishing trips annually. [ABSTRACT FROM AUTHOR]

In this study, we develop a mathematical model to examine the dynamics of COVID-19 with diabetes. Recognizing the increased vulnerability of diabetic patients, we proposed separate isolation classes for COVID-19 cases with and without diabetes. The model is parameterized using real data from India for the period March 2021 to September 2021. Sensitivity analysis for the basic reproduction number shows that isolation (for COVID-19 with diabetes) plays a significant role in lessening COVID-19 cases. Through numerical evaluations, we demonstrate that timely and focused care, including hospitalization/isolation, and treatment for COVID-19 patients with diabetes, can substantially reduce the disease burden by nearly three-fold. The optimal control problem with different strategies has also been discussed. We obtain that vaccination (with the least Average Cost-Effectiveness Ratio (ACER) and a combined strategy (with the highest Infection-Averted Ratio (IAR) are the most cost-saving and effective interventions to eradicate the disease. [ABSTRACT FROM AUTHOR]

The regularization of spacecraft motion equations by the Kustaanheimo–Stiefel transformation for coordinates and Sundman's transformation for time is considered in the problem of low-thrust optimal transfer. From Pontryagin's maximum principle, the thrust vector optimal control is derived under the limited power condition. The Earth–Mars transfer problem is solved in regular variables. The comparison of calculated trajectories with the ones obtained by the parameter continuation method is performed, and the stability properties of the two-point boundary value problem in the Cartesian and regular variables are studied. [ABSTRACT FROM AUTHOR]

In the dynamic field of inventory management, effective control of perishable goods presents a significant challenge. Additionally, most organizations are greatly concerned about the bullwhip effect, which is further exacerbated in the case of perishable goods. This paper considers an effective control theoretic analysis of the periodic review strategy for perishable production-inventory systems. In contrast to the earlier approaches, which are mainly based on heuristics and static optimization, discrete Pontryagin’s maximum principle of dynamic optimization is applied for the analytical solution of the problem. In the proposed model, the time-variant deteriorating stock is replenished with delay from its manufacturing unit in order to fulfil unknown stock-dependent time-varying demand. Handling loss in the saturating integrator is considered, but handling time is neglected. The inventory model is framed as a linear quadratic regulator to reduce the risk of the bullwhip effect. The other benefit is to get the optimal solution analytically. The supply of goods from its manufacturing unit also helps minimize the bullwhip effect. The optimal production and optimal growth of the inventory are calculated based on some specific choice of the parametric functions involved in the process, and they are represented graphically. The problem of periodic blood supply to the Blood bank through blood donation camps is studied as a practical application of the proposed inventory model. Finally, the proposed model works well for real-world scenarios under sensitivity analysis with a proper choice of parameters. [ABSTRACT FROM AUTHOR]

A way of determining the approximate boundary of a reachable set is proposed for a class of nonsmooth controlled dynamic systems on a plane in economics. The technique is based on an explicit procedure for smoothing the system and using the apparatus of Pontryagin's maximum principle. As an example, the problem is posed of constructing the boundary of the reachable set for a controlled version of the well-known Kaldor business cycle model. [ABSTRACT FROM AUTHOR]

Fuel consumption is one of the main concerns for heavy-duty trucks. Predictive cruise control (PCC) provides an intriguing opportunity to reduce fuel consumption by using the upcoming road information. In this study, a real-time implementable PCC, which simultaneously optimizes engine torque and gear shifting, is proposed for heavy-duty trucks. To minimize fuel consumption, the problem of the PCC is formulated as a nonlinear model predictive control (MPC), in which the upcoming road elevation information is used. Finding the solution of the nonlinear MPC is time consuming; thus, a real-time implementable solver is developed based on Pontryagin's maximum principle and indirect shooting method. Dynamic programming (DP) algorithm, as a global optimization algorithm, is used as a performance benchmark for the proposed solver. Simulation, hardware-in-the-loop and real-truck experiments are conducted to verify the performance of the proposed controller. The results demonstrate that the MPC-based solution performs nearly as well as the DP-based solution, with less than 1% deviation for testing roads. Moreover, the proposed co-optimization controller is implementable in a real-truck, and the proposed MPC-based PCC algorithm achieves a fuel-saving rate of 7.9% without compromising the truck's travel time. [ABSTRACT FROM AUTHOR]

Single-use anion-exchange resins can reduce hazardous chromates to safe levels in drinking water. However, since most process control strategies monitor effluent concentrations, detection of any chromate leakage leads to premature resin replacement. Furthermore, variations in the inlet chromate concentration and other process conditions make process control a challenging step. In this work, we capture the uncertainty of the process conditions by applying the Ito process of Brownian motion with a drift into a stochastic optimal control strategy. The ion-exchange process is modeled using the method of moments, which helps capture the process dynamics, later formulated into mathematical objectives representing desired chromate removal. We then solved our developed models as an optimal control problem via Pontryagin's maximum principle. The objectives enabled a successful control via flow rate adjustments leading to higher chromate extraction. Such an approach maximizes the capacity of the resin and column efficiency to remove toxic compounds from water while capturing deviations in the process conditions. When dealing with highly toxic compounds like chromium, it is critical that its concentration in drinking water is kept at a low, safe level. As single-use ion-exchange resins are used to extract the hazardous chemical, changes in inlet concentrations can lead to premature leakage. Hence, an optimal control strategy is needed for the purification system while monitoring the inlet concentration rather than the outlet concentration to avoid a process control delay. For a successful optimization, predicting the output concentration based on the inlet conditions becomes necessary to maximize the performance of the extraction process. In this work, predictive modeling while capturing the uncertainties of the system maximized the chromate removal in less time than running the process at a constant flow rate. The results demonstrate that changing the flow rate with time is an improved strategy to achieve such performance. The flow rate change is a unique approach to an industry that designs its processes around a constant flow rate and reacts too late when system deviations have already occurred. Therefore, applying the approach described in this work will maximize the utilization of the purification process resulting in less waste produced and safer drinking water. [ABSTRACT FROM AUTHOR]

A scam rumor is a false information that is spread in order to deceive individuals and organizations. These rumors frequently use people’s trust and emotions for illegal purposes to gain money or cause harm. This paper presents an optimal control strategy and a cost effectiveness analysis for a deterministic model designed to explain how false information, or ”Scam Rumor” spreads on social networking sites. This model accounts for the network’s informants’ actions. We evaluate the efficacy of tactics to stop the propagation of scam rumors via social media using optimal control theory. We also do numerical simulations to evaluate the benefits and drawbacks of applying the techniques we suggest, emphasizing their potential to improve the accuracy of online content.

According to the World Health Organization’s guidelines, a two-year mandatory breastfeeding period is strongly recommended, supported by research demonstrating its significant benefits in reducing childhood illnesses and mortality rates. The present study is designed to address this recommendation by focusing on a specific infection, namely rotavirus. Acknowledging this evidence, researchers recognize the need for a deeper understanding of the dynamics of rotavirus disease. In response, our study focuses on constructing and analyzing an SIR epidemic model uniquely designed for transmitting rotavirus in children. The model partitions the children population into three compartments: susceptible, infected, and recovered. To enhance precision, we further categorize susceptible children based on breastfeeding status and infected individuals based on infectious or non-infectious nature. This detailed categorization enables a thorough examination of rotavirus transmission dynamics. The proposed model analyzes two equilibria, disease-free and endemic, revealing local and global stability conditions determined by the basic reproduction number (R0). Our exploration extends to local stability analysis for the endemic equilibrium when R0>1 and global stability assessment using the Lyapunov theory under specific conditions. Subsequently, we employ optimal control theory through Pontryagin’s maximum principle to minimize the cost burden associated with disease control in children. Then, our work provides a sensitivity analysis of different parameters for the basic reproduction number to enhance our understanding of model dynamics. Finally, numerical simulations conducted with MATLAB and Python software validate our analytical findings, offering a comprehensive and practical assessment of the proposed model’s implications for controlling rotavirus transmission in children.

The problems of optimizing the trajectory motion of the first stage of an aerospace system according to the criterion of the maximum of the final mass are considered. The control is the angle of attack and thrust of the engines. Control optimization is carried out on the trajectory section from the point of bringing the first stage to the launch area until the motion parameters required for separation of the space stage are reached. The Pontryagin’s maximum principle is used to determine optimal control programs. The solution of the problem without restrictions on the modes of motion is carried out using the example of acceleration and climb of the first stage of the RASCAL aerospace system. A method is proposed for determining approximate optimal control in a problem with a limitation on the altitude range of the engines with separate optimization of the active and passive sections and the search for the optimal point of their coupling. Changes in control program, trajectory, and fuel consumption are discussed when limiting the maximum flight altitude in the active section.

In pursuit–evasion scenarios, the pursuer typically possesses a lethal zone. If the evader effectively utilizes perceptual information, they can narrowly escape the lethal zone while minimizing energy consumption, thereby avoiding excessive and unnecessary maneuvers. Based on optimal control theory, we propose a guidance law for achieving critical safe miss distance evasion under bounded control. First, we establish the zero-effort miss (ZEM) state equation for the evader, while approximating disturbances from the pursuer. Next, we formulate an optimal control problem with energy consumption as the objective function and the ZEM at the terminal time as the terminal constraint. Subsequently, we design an iterative algorithm that combines the homotopy method and Newton’s iteration to solve the optimal control problem, applying Pontryagin’s Maximum Principle. The simulation results indicate that the designed iterative method converges effectively; through online updates, the proposed guidance law can successfully achieve critical safe miss distance evasion. Compared to programmatic maneuvering and norm differential game guidance law, this approach not only stabilizes the evader’s evasion capabilities but also significantly reduces energy consumption.

This research article presents a mathematical model that tracks and monitors the spread of COVID-19 strains in a discrete time frame. The study incorporates two control strategies to reduce the transmission of these strains: vaccination and providing appropriate treatment and medication for each strain separately. Optimal controls were established using Pontryagin's maximum principle in discrete time, and the optimality system was solved using an iterative method. To validate the effectiveness of the theoretical findings, numerical simulations were conducted to demonstrate the impact of the implemented strategies in limiting the spread of COVID-19 mutant strains. [ABSTRACT FROM AUTHOR]

In this paper, a simple and highly accurate approximate method for solving optimal control problems (OCPs) is presented. In this method, initially, by using the necessary optimality conditions based on the Pontryagin’s maximum principle, the given OCP is transformed into a two­point boundary value problem (BVP). Then, by applying a second­order finite difference formula, the resulting BVP is discretized, and a system of algebraic equations is formulated. Convergence analysis of the proposed method is also discussed, and matrix formulations are provided for ease of implementation. The numerical results obtained in this study are compared with some other methods, demonstrating the high accuracy, speed, and efficiency of the proposed method for solving both linear and nonlinear OCPs. [ABSTRACT FROM AUTHOR]

In this paper, we study the modern mathematical theory of the optimal control problem associated with the fractional Roesser model and described by Caputo partial derivatives, where the functional is given by the Riemann-Liouville fractional integral. In the formulated problem, a new version of the increment method is applied, which uses the concept of an adjoint integral equation. Using the Banach fixed point principle, we prove the existence and uniqueness of a solution to the adjoint problem. Then the necessary and sufficient optimality condition is derived in the form of the Pontryagin's maximum principle. Finally, the result obtained is illustrated by a concrete example. [ABSTRACT FROM AUTHOR]