Maximum Principle Research Articles

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.

Qualitative analysis of fourth-order hyperbolic equations

We investigate the qualitative properties of weak solutions to the boundary value problems for fourth-order linear hyperbolic equations with constant coefficients in a plane bounded domain convex with respect to characteristics. Our main scope is to prove some analog of the maximum principle, solvability, uniqueness and regularity results for weak solutions of initial and boundary value problems in the space L2. The main novelty of this paper is to establish some analog of the maximum principle for fourth-order hyperbolic equations. This question is very important due to natural physical interpretation and helps to establish the qualitative properties for solutions (uniqueness and existence results for weak solutions). The challenge to prove the maximum principle for weak solutions remains more complicated and at that time becomes more interesting in the case of fourth-order hyperbolic equations, especially, in the case of non-classical boundary value problems with data of weak regularity. Unlike second-order equations, qualitative analysis of solutions to fourth-order equations is not a trivial problem, since not only a solution is involved in boundary or initial conditions, but also its high- order derivatives. Other difficulty concerns the concept of weak solution of the boundary value problems with L2 – data. Such solutions do not have usual traces, thus, we have to use a special notion for traces to poss correctly the boundary value problems. This notion is traces associated with operator L or L-traces. We also derive an interesting interpretation (as periodicity of characteristic billiard or the John's mapping) of the Fredholm's property violation. Finally, we discuss some potential challenges in applying the results and proposed methods.

Maximum principle for partially observed risk-sensitive optimal control problem of McKean–Vlasov FBSDEs involving impulse controls

Maximum principle for partially observed risk-sensitive optimal control problem of McKean–Vlasov FBSDEs involving impulse controls

An approach for regime-switching stochastic control problems with memory and terminal conditions

In this research article, we focus on a stochastic optimal control problem with two types of terminal constraints. These specific conditions provide real-valued and stochastic Lagrange multipliers. Our model evolves according to a Markov regime-switching jump diffusion model with memory. In this context, the memory is represented by a Stochastic Differential Delay Equation. We present two theorems for each constraint within the general formulation of stochastic optimal control theory in a Lagrangian environment. We approach to this task from a theoretical perspective and provide mild technical assumptions, which make our theorems applicable for a broad class of stochastic control problems as well as for a wide range of disciplines such as engineering, biology, operations research, medicine, computer science and economics. In this work, we apply Stochastic Maximum Principle to demonstrate an optimal dividend policy corresponding to a time-delayed wealth process of a company. Moreover, we determine the real-valued Lagrange multiplier of this control problem explicitly.

A GUMBORO-SALMONELLA CO-INFECTION MATHEMATICAL MODEL WITH OPTIMAL CONTROL

Poultry production contributes immensely to the economic growth of a country. For instance in Kenya, 20tonnes of poultry meat worth 3.5 billion kenya shillings and 1.3 billion eggs worth 9.7 billion Kenya Shillingscome from this sector. However, the sector is greatly threatened by poultry diseases among them, Gum-boro(IBD) and Salmonella as inadequate knowledge exists of optimal control strategies for various poultryco-infections. In this research, a Gumboro-Salmonella co-infection mathematical model with optimal controlis developed using a system of ODEs to perform an optimal control analysis. The analysis was done by for-mulating an optimal control problem and using Pontryagin’s maximum principle to solve it. The Numericalsimulation results showed that the best Gumboro-Salmonella co-infection control strategy involved combiningall the interventions

Point-feature label placement with maximum entropy principle

A map becomes readable and translatable only after the use of labels. High-quality label placement (i.e., labelling) is a combinatorial optimization problem, where one or more objective functions are required. However, such objective functions are still not well achieved in existing methods with commonly used labelling rules (e.g., “avoidance of overlapping labels” and “placement at priority positions”). In this study, we define a new objective function using the maximum entropy principle and employ a modified genetic algorithm in the optimization process. Comparative experiments have been conducted with 29 existing labelling methods in a dataset of 1000 points. Experimental results show that the average percentage of non-overlapping labels by this new method reaches 94.73% (i.e., being ranked second in all 30 methods) and has no statistically significant difference from the best one (i.e., 94.87%). By further analysis, it is found that this new method not only can place more labels in priority positions than the method being ranked first but also has the potential to place labels for line and area features. Such results indicate that the maximum entropy principle may pave the way for further research in cartography.

Fundamental solution, maximum principle and Harnack inequality for second order subelliptic operators

Fundamental solution, maximum principle and Harnack inequality for second order subelliptic operators

Transmission Dynamics of Typhoid Fever Outbreak: A Mathematical Modelling and Optimal Control Approach

In this paper, we propose and analyze a deterministic mathematical model for the control of typhoid fever. This model incorporates five effective control strategies, including vaccination, booster vaccination, personal hygiene, treatment, and bacteria sterilization, in order to effectively stop the spread of the disease in the population. The model consists of seven compartments, each representing a different stage in the disease’s progression: vaccinated, susceptible, carrier, infected, recovered, booster vaccinated, and bacterial. Using the next-generation matrix approach, we calculated the reproduction number , which measures the disease’s potential for spreading. Our stability analysis, using the Routh-Hurwitz criteria, revealed that the disease-free equilibrium point is locally asymptotically stable when is less than 1, among other conditions. This means that typhoid can be completely eradicated if the rate of secondary infection is kept below a certain threshold. Further sensitivity analyses were conducted to determine the key parameters that impact . Based on our findings, we formulated an optimal control problem and utilized Pontryagin’s Maximum Principle to determine the most effective strategy for controlling and eliminating the disease. Our results show that a combination of vaccination, booster vaccination, personal hygiene, treatment, and bacteria sterilization is the most optimal approach. With our comprehensive model and effective control strategies, we are one step closer to wiping out typhoid fever for good.

Optimal seasonal schedule for producing biogenic volatile organic compounds for tree defense

The leaves of many trees emit biogenic volatile organic compounds (BVOCs) that protect them from various threats, including herbivory, pathogens, and heat stress. In a previous study, we analyzed the optimal seasonal schedule for producing isoprene, a highly volatile BVOC, in leaves to mitigate heat damage and maximize net carbon gain. In this paper, we investigate the seasonal production schedule of BVOCs stored in leaves, such as monoterpenes and sesquiterpenes, which decay slowly. When the leaves are bitten, these chemicals are emitted and help to prevent further herbivory. The optimal seasonal schedule, analyzed using Pontryagin’s maximum principle, includes a period of singular control. Producing BVOCs for defense is advantageous if their decay rate is slow and the photosynthetic rate is fast. The amount of BVOCs produced increases with slower decay rate and faster photosynthetic rate. But it does not increase monotonically with the magnitude of the threat. BVOCs are produced earlier than the peak period of the threat for which the chemicals are intended. Based on the results of the model, we discuss the reported variations in BVOC production among different chemical species and tree species, as well as the seasonal patterns of gene expression in different pathways for BVOC production.

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.

Existence of ground state sign-changing solutions for a class of quasilinear scalar field equations originating from nonlinear optics

This paper is mainly concerned with the existence of ground state sign-changing solutions for a class of second order quasilinear elliptic equations in bounded domains, which derived from nonlinear optics models. Combining a non-Nehari manifold method due to Tang and Cheng [J. Differ. Equations 261, 2384–2402 (2016)] and a quantitative deformation lemma with Miranda theorem, we obtain that the problem has at least one ground state solution and one ground state sign-changing solution with two precise nodal domains and we present the concrete lower bounds of ground state solution and ground state energy. We also obtain that any weak solution of the problem has C1,σ-regularity for some σ ∈ (0, 1). With the help of Maximum Principle, we reach the conclusion that the energy of the ground state sign-changing solutions is strictly larger than twice that of the ground state solutions.

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.

Bilinear optimal control for a fractional diffusive equation

We consider a bilinear optimal control for an evolution equation involving the fractional Laplace operator of order 0<s<1. We first give some existence and uniqueness results for the considered evolution equation. Next, we establish some weak maximum principle results allowing us to obtain more regularity of our state equation. Then, we consider an optimal control problem which consists to bring the state of the system at final time to a desired state. We show that this optimal control problem has a solution and we derive the first- and second-order optimality conditions.

An asymptotic-preserving and exactly mass-conservative semi-implicit scheme for weakly compressible flows based on compatible finite elements

We present a novel asymptotic-preserving semi-implicit finite element method for weakly compressible and incompressible flows based on compatible finite element spaces. The momentum is sought in an H(div)-conforming space, ensuring exact pointwise mass conservation at the discrete level. We use an explicit discontinuous Galerkin-based discretization for the nonlinear convective terms, while treating the pressure and viscous terms implicitly, so that the CFL condition depends only on the fluid velocity. To handle shocks and damp spurious oscillations in the compressible regime, we incorporate an a posteriori limiter that employs artificial viscosity and is based on a discrete maximum principle. By using hybridization, the final algorithm requires solving only symmetric positive definite linear systems. As the Mach number approaches zero and the density remains constant, the method naturally converges to an H(div)-based discretization of the incompressible Navier-Stokes equations in the vorticity-velocity-pressure formulation. Several numerical tests validate the proposed method.

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.

Stochastic optimal control of pre-exposure prophylaxis for HIV infection for a jump model.

We analyze a stochastic optimal control problem for the PReP vaccine in a model for the spread of HIV. To do so, we use a stochastic model for HIV/AIDS with PReP, where we include jumps in the model. This generalizes previous works in the field. First, we prove that there exists a positive, unique, global solution to the system of stochastic differential equations which makes up the model. Further, we introduce a stochastic control problem for dynamically choosing an optimal percentage of the population to receive PReP. By using the stochastic maximum principle, we derive an explicit expression for the stochastic optimal control. Furthermore, via a generalized Lagrange multiplier method in combination with the stochastic maximum principle, we study two types of budget constraints. We illustrate the results by numerical examples, both in the fixed control case and in the stochastic control case.

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.

Global Analysis of Meningitis Disease With Optimal Control

The meningitis epidemic has impacted lives negatively, especially in Sub-Sahara Africa, dubbed the ‘Meningitis Belt’. The epidemic has been a public health concern due to an improper understanding of the disease’s dynamics. To implement a control measure that will help minimize the epidemic, we introduce a non-linear Meningitis model that describes the dynamic behaviour of the disease and explains the transmission trend. The model explores the condition that leads to local or global asymptomatic stability of the equilibria. The model is subjected to a sensitivity analysis to find the parameters that influence the R0. The model is modified into an optimal control by adding timedependent controls. The control model is solved qualitatively using Pontryagin’s maximum principle and numerically using MATLAB and the fourth-order Runge-Kutta method. We provide a control strategy that can be relied on for management decision-making based on the results.

Finite Horizon Stochastic H2/H∞ Control for Continuous‐Time Systems Driven by G‐Brownian Motion

ABSTRACTIn this article, we consider a stochastic control problem under volatility ambiguity. In this framework, the state function is described by a stochastic differential equation driven by ‐Brownian motion. Different from the classical approach, the volatility ambiguity is characterized by a family of non‐dominated probability measures. control can be modeled as a “inf‐sup problem” and control can be modeled as a “sup‐sup problem.” With the help of stochastic maximum principle and dynamic programming principle, we derive the stochastic bounded real lemma and two sets of cross coupled generalized differential Riccati equations, which is associated with the solvability of control.

The relationship between maximum principle and dynamic programming principle for stochastic recursive control problem with random coefficients

The relationship between maximum principle and dynamic programming principle for stochastic recursive control problem with random coefficients