Maximum Principle Research Articles

A novel grade assessment method for cybersecurity situation of online retailing with decision makers’ bounded rationality

The online retailing cybersecurity situations grade assessment (GS) is a key issue in cybersecurity management, which can be regarded as a type of multi-attributes GS problems. However, traditional GS methods rarely discuss the boundary fuzziness and hesitation in the grade classification of attributes, as well as the monotony relation and interrelation between entropy fuzziness and intuitionism. A novel GS method with respect to decision makers’ bounded rationality is proposed. To get the method, the membership functions (MFs) and non-membership functions (NMFs) of attributes’ eigenvalue grade considering decision maker s’ bounded rationality and hesitation are proposed. An axiomatic definition of IF entropy on the basis of isentropic arcs is proposed, providing the formula of the monotony relation and interrelation between fuzziness and intuitionism. We, considering the reliability and the IF grade assessment method, propose the similarity of IF grade eigenvalue, which identifies grades reasonably and distinguishes the ranking of the same grade, and overcomes the defect that the maximum membership degree principle ignores the position differences of each grade assessment level.

A finite element analysis study on different angle correction designs for inclined implants in All-On-Four protocol

BackgroundThe aim of this study is to investigate, through finite element analysis (FEA), the biomechanical behavior of the built-in angle corrected dental implant versus implant with angled multiunit abutment used in All-On-Four treatment protocol.MethodsTwo (3D) finite element models of a simplified edentulous mandible were constructed with two different posterior implant designs based on the All-On-Four protocol. Four implants were placed in each model, the two anterior implants were positioned vertically at the lateral incisor/canine sites. Depending on the implant fixture design in posterior area, there are two models created; Model I; the mandible was rehabilitated with four co-axis (4 mm in diameter × 15 mm in length) implants with distally built-in angle corrected implants (24-degree angle correction) .While Model II, the mandible was rehabilitated with four conventional (4 mm in diameter × 14 mm in length) implants with a distally inclined posterior implants (25 degree) and angled multiunit abutments. CAD software (Solidworks© 2017; Dassault Systems Solidworks Corp) was used to model the desired geometry. Axial and inclined Loads were applied on the two models. A Finite element analysis study was done using an efficient software ANSYS© with specified materials. The resultant equivalent Von-Misses stresses (VMS), maximum principal stresses and deformation analysis were calculated for each part (implants and prosthetic components).ResultsWhen applying axial and non-axial forces, model II (angled multiunit model) showed higher deformation on the level of Ti mesh about 13.286 μm and higher VMS 246.68 MPa than model I (angle corrected implant). Model I exhibited higher maximum stresses 107.83 MPa than Model II 94.988 MPa but the difference was not statistically significant.ConclusionWithin the limitation of the FEA study, although angle correcting implant design is showing higher values in maximum principle stresses compared with angled multiunit abutments, model deformation and resultant VMS increased with angled multiunit abutments. The angle correcting designs at implant level have more promising results in terms of deformation and VMS distribution than angle correction at abutment level.

Sliding methods for dual fractional nonlinear divergence type parabolic equations and the Gibbons’ conjecture

Abstract In this paper, we consider the general dual fractional parabolic problem ∂ t α u ( x , t ) + L u ( x , t ) = f ( t , u ( x , t ) ) in R n × R . ${\partial }_{t}^{\alpha }u\left(x,t\right)+\mathcal{L}u\left(x,t\right)=f\left(t,u\left(x,t\right)\right) \text{in} {\mathbb{R}}^{n}{\times}\mathbb{R}.$ We show that the bounded entire solution u satisfying certain one-direction asymptotic assumptions must be monotone increasing and one-dimensional symmetric along that direction under an appropriate decreasing condition on f. Our result here actually solves a well-known problem known as Gibbons’ conjecture in the setting of the dual fractional parabolic equations. To overcome the difficulties caused by the nonlocal divergence type operator L $\mathcal{L}$ and the Marchaud time derivative ∂ t α ${\partial }_{t}^{\alpha }$ , we introduce several new ideas. First, we derive a general weighted average inequality corresponding to the nonlocal operator L $\mathcal{L}$ , which plays a fundamental bridging role in proving maximum principles in unbounded domains. Then we combine these two essential ingredients to carry out the sliding method to establish the Gibbons’ conjecture. It is worth noting that our results are novel even for a special case of L $\mathcal{L}$ , the fractional Laplacian (−Δ) s , and the approach developed in this paper will be adapted to a broad range of nonlocal parabolic equations involving more general Marchaud time derivatives and more general non-local elliptic operators.

An energy management strategy for fuel cell hybrid electric vehicle based on a real-time model predictive control and pontryagin’s maximum principle

ABSTRACT 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.

Constrained Reweighting of Distributions: An Optimal Transport Approach.

We commonly encounter the problem of identifying an optimally weight-adjusted version of the empirical distribution of observed data, adhering to predefined constraints on the weights. Such constraints often manifest as restrictions on the moments, tail behavior, shapes, number of modes, etc., of the resulting weight-adjusted empirical distribution. In this article, we substantially enhance the flexibility of such a methodology by introducing a nonparametrically imbued distributional constraint on the weights and developing a general framework leveraging the maximum entropy principle and tools from optimal transport. The key idea is to ensure that the maximum entropy weight-adjusted empirical distribution of the observed data is close to a pre-specified probability distribution in terms of the optimal transport metric, while allowing for subtle departures. The proposed scheme for the re-weighting of observations subject to constraints is reminiscent of the empirical likelihood and related ideas, but offers greater flexibility in applications where parametric distribution-guided constraints arise naturally. The versatility of the proposed framework is demonstrated in the context of three disparate applications where data re-weighting is warranted to satisfy side constraints on the optimization problem at the heart of the statistical task-namely, portfolio allocation, semi-parametric inference for complex surveys, and ensuring algorithmic fairness in machine learning algorithms.

Optimality conditions for bilevel optimal control problems with non-convex quasi-variational inequalities

We establish Pontryagin optimality conditions for a generalized bilevel optimal control problem in which the leader is subject to a pure state inequality constraint, while the follower is governed by a non-convex quasi-variational inequality parameterized by the final state. To simplify the problem at hand, we convert it into a single-level optimal control problem by mapping the solution set of the quasi-variational inequality to a parametric optimization problem and employing the value function reformulation. Furthermore, we introduce certain regularity conditions to ensure that the derived maximum principle remains non-degenerate. Finally, we provide an illustrative example to elucidate our research findings.

Optimal control of susceptible mature pest concerning disease-induced pest-natural enemy system with cost-effectiveness

Abstract This article addresses the pressing issue of pest outbreaks in India, which poses significant challenges for farmers and ecologists. A novel system is proposed for effective control that leverages natural enemies. Here, the pests are classified into juveniles and mature individuals, further categorized as susceptible or infected. The study introduces harvesting, incorporating external efforts and natural phenomena, in a pest-epidemic prey–predator system featuring a prey-stage structure. The model reveals three equilibria: trivial, boundary (indicating the absence of natural enemies), and interior equilibria. Notably, the trivial equilibrium is consistently unstable. As demonstrated by stability analysis, the survival or extinction of natural enemies hinges on control variables, including the harvesting rate, disease transmission rate, and natural death rate. Local stability is assessed using the Routh–Hurwitz criterion, while global stability is explored through the Lyapunov method. Furthermore, optimal control theory and Pontryagin’s maximum principle are applied for model optimization, unveiling crucial optimality conditions and determining the optimal harvesting rate for susceptible mature prey. Numerical computations validate theoretical insights, offering valuable guidance for formulating policies that optimize the control of susceptible adult pests within a disease-induced pest-natural enemy system, ensuring sustained cost-effectiveness.

Optimal control and bifurcation analysis of SEIHR model for COVID-19 with vaccination strategies and mask efficiency

Abstract In this article, we present a susceptible, exposed, infected, hospitalized and recovered compartmental model for COVID-19 with vaccination strategies and mask efficiency. Initially, we established the positivity and boundedness of the solutions to ensure realistic predictions. To assess the epidemiological relevance of the system, an examination is conducted to ascertain the local stability of the endemic equilibrium and the global stability across two equilibrium points are carried out. The global stability of the system is demonstrated using Lyapunov’s direct method. The disease-free equilibrium is globally asymptotically stable when the basic reproduction number (BRN) is less than one, whereas the endemic equilibrium is globally asymptotically stable when BRN is greater than one. A sensitivity analysis is performed to identify the influential factors in the BRN. The impact of various time-dependent strategies for managing and regulating the dynamic transmission of COVID-19 is investigated. In this study, Pontryagin’s maximum principle for optimal control analysis is used to identify the most effective strategy for controlling the disease, including single, coupled, and threefold interventions. Single-control interventions reveal physical distancing as the most effective strategy, coupled measures reduce exposed populations, and implementing all controls reduces susceptibility and infections.

A boundary maximum principle for stationary pairs of varifolds with fixed contact angle

Abstract In this note, we establish a boundary maximum principle for a class of stationary pairs of varifolds satisfying a fixed contact angle condition in any compact Riemannian manifold with smooth boundary.

Mathematical Model and Optimal Control of Covid-19 in Nigeria

The global pandemic, Covid-19, caused by corona virus disease is responsible for a significant number of deaths and huge economic losses in almost all the countries of the world, including Nigeria. In order to manage the spread of this disease in Nigeria, the Nigeria Center for Disease Control(NCDC) has proposed and implemented various control and preventive measures such as vaccination, use of alcohol-based hand sanitizers, social distancing, and others. The aim of this paper is to model the transmission dynamics of Covid-19 in Nigeria, and obtain, by using Pontryagin Maximum Principle, the combination of these control strategies for effective control of the disease in Nigeria. Numerical experiments with Nigeria Covid-19 data show the effectiveness optimal use of these preventive and control measures for Covid-19.

Impacts of optimal control strategies on the HBV and COVID-19 co-epidemic spreading dynamics

Different cross-sectional and clinical research studies investigated that chronic HBV infected individuals’ co-epidemic with COVID-19 infection will have more complicated liver infection than HBV infected individuals in the absence of COVID-19 infection. The main objective of this study is to investigate the optimal impacts of four time dependent control strategies on the HBV and COVID-19 co-epidemic transmission using compartmental modeling approach. The qualitative analyses of the model investigated the model solutions non-negativity and boundedness, calculated all the models effective reproduction numbers by applying the next generation operator approach, computed all the models disease-free equilibrium point (s) and endemic equilibrium point (s) and proved their local stability, shown the phenomenon of backward bifurcation by applying the Center Manifold criteria. By applied the Pontryagin’s Maximum principle, the study re-formulated and analyzed the co-epidemic model optimal control problem by incorporating four time dependent controlling variables. The study also carried out numerical simulations to verify the model qualitative results and to investigate the optimal impacts of the proposed optimal control strategies. The main finding of the study reveals that implementation of protections, COVID-19 vaccine, and treatment strategies simultaneously is the most effective optimal control strategy to tackle the HBV and COVID-19 co-epidemic spreading in the community.

FCE-based health status evaluation and IGM-based failure prediction of tool magazine manipulator for CNC machine tools

The reliability of machine tool components, particularly the tool magazine manipulator, significantly affects the overall performance of the machine tool. To address the challenge of accurately evaluating the manipulator’s health status using a single performance indicator, this study proposes a method that combines Fuzzy Comprehensive Evaluation (FCE) and a Combined Weighting Method (CWM). By considering both subjective and objective factors, this method provides a comprehensive evaluation of the manipulator’s health status, enhancing the accuracy and reliability of the assessment. The method utilizes fuzzy distribution to construct membership matrices for different health levels and adopts the CWM that combines the Entropy Weight Method (EWM) and Analytic Hierarchy Process (AHP) to determine the combined weights of the health evaluation indices. This approach improves the accuracy and reliability by considering multiple indicators and objectively weighting them based on their importance. The current health status of the manipulator is evaluated using the fuzzy weighted average operator and the maximum membership principle. Moreover, a fault prediction method based on Particle Swarm Optimization (PSO) and GM(1,1) is proposed to overcome the information gap and small sample problems. The proposed model’s prediction accuracy is verified by comparing it with other models, demonstrating its effectiveness and reliability.

Global dynamics and optimal control of a nonlinear fractional-order cholera model

In this article, a fractional-order epidemic model for cholera is proposed and analyzed. Two transmission routes for cholera are considered to develop the compartmental epidemic model. The basic biological properties of the solutions of the fractional-order model are investigated. The global asymptotic stability of the equilibrium points have been established using appropriate Lyapunov functional. Moreover, a fractional-order control problem is presented, and its analytical solution is derived using Pontryagin’s maximum principle. Also, some graphical visualizations of the theoretical results are provided. It is found that the factional-order derivative only affect the time to reach the stationary states. Sensitivity analysis reveals that by reducing the rates of new recruitment and both the disease transmission rates, it may be possible to reduce the value of the basic reproduction number.

The non-holonomic Herglotz variational problem

The geometric approach to the study of the Herglotz problem developed in Massa and Pagani [J. Math. Phys. 64, 102902 (2023)] is extended to the case in which the evolution of the system is subject to a set of non-holonomic constraints. The original setup is suitably adapted to the case in study. Various aspects of the problem are considered: the direct derivation of the evolution equations; the super-lagrangian approach; the resulting super-Hamiltonian and its relation with Pontryagin’s maximum principle; the abnormality index of the extremals; the invariance properties of the theory and the consequent existence of Herglotz Lagrangians gauge equivalent to ordinary ones.

Analysis of a class of completely non-local elliptic diffusion operators

This work explores the possibility of developing the analog of some classic results from elliptic PDEs for a class of fractional ODEs involving the composition of both left- and right-sided Riemann-Liouville (R-L) fractional derivatives, Da+αDb-β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${D^\\alpha _{a+}}{D^\\beta _{b-}}$$\\end{document}, 1<α+β<2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1<\\alpha +\\beta <2$$\\end{document}. Compared to one-sided non-local R-L derivatives, these composite operators are completely non-local, which means that the evaluation of Da+αDb-βu(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${D^\\alpha _{a+}}{D^\\beta _{b-}}u(x)$$\\end{document} at a point x will have to retrieve the information not only to the left of x all the way to the left boundary but also to the right up to the right boundary, simultaneously. Therefore, only limited tools can be applied to such a situation, which is the most challenging part of the work. To overcome this, we do the analysis from a non-traditional perspective and eventually establish elliptic-type results, including Hopf’s Lemma and maximum principles. As α→1-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\rightarrow 1^-$$\\end{document} or α,β→1-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha ,\\beta \\rightarrow 1^-$$\\end{document}, those operators reduce to the one-sided fractional diffusion operator and the classic diffusion operator, respectively. For these reasons, we still refer to them as “elliptic diffusion operators", however, without any physical interpretation.

Error Estimates in the Maximum Norm for the Solution of Poisson’s Equation Approximated by the Five-Point Laplacian Using the Discrete Maximum Principle

In this paper, we study error estimates in the maximum norm in the context of solving Poisson’s equation numerically when approximated the Five-Point Laplacian method using the discrete maximum principle. The primary objective is to assess the accuracy of this numerical approach in solving Poisson’s equation and to provide insights into the behavior of error estimates. We focus on the estimates of maximum norm of the discrete functions defined on a grid in a unit square as well as in a square of side s, and estimate errors measured in the maximum norm.

GLOBAL SENSITIVITY ANALYSIS AND OPTIMAL CONTROL OF TYPHOID FEVER TRANSMISSION DYNAMICS

This paper presents a mathematical model aimed at studying the global behaviour and optimal control strategies for Typhoid fever. The primary objective of this study is to identify the most effective control strategy that minimizes the spread of the disease. To achieve this, we calculate the effective and basic reproduction numbers and utilize them to investigate the existence and stability of the equilibria. Furthermore, we investigate the global impact of each model parameter on the variables using Latin Hypercube Sampling and Partial Rank Correlation Coefficient. The necessary conditions of the optimal control problem are analyzed using Pontryagin’s maximum principle, and the numerical values of the model parameters are estimated using the maximum likelihood estimator. The results indicate that the optimal use of vaccination for susceptible individuals, as well as the screening and treatment of asymptomatic infected individuals, have a significant impact on reducing the spread of the disease in endemic regions.

Upper and lower solutions method for a class of second-order coupled systems

This paper provides a class of upper and lower solution definitions for second-order coupled systems by transforming the fourth-order differential equation into a second-order differential system. Then, by constructing a homotopy parameter and utilizing the maximum principle, we propose an upper and lower solutions method for studying a class of second-order coupled systems with Dirichlet boundary conditions and obtain an existence result.

3D numerical simulation of an anisotropic bead type thermistor and multiplicity of solutions

We perform some 3D numerical experiments for the approximation of the solutions to a bead type thermistor problem. We consider the case of a diagonal anisotropic diffusion matrix whose jth entry is of the form |∂u/∂xj|pj−2∂u/∂xj, u being the temperature inside the thermistor and the exponents pj, 1≤j≤3, lie in the interval (1,+∞). We first show some existence results for different notions of solutions, prove a maximum principle for each type of solution, and study certain symmetry properties for these solutions in a bead type thermistor. These properties lead us to the introduction of a symmetric solution and we show the existence of such a solution.We have developed a numerical algorithm for the computation of the numerical solutions in a bead type thermistor. This algorithm combines a fixed-point technique with a standard finite element method (FEM). Some numerical tests have shown the existence of non-symmetric solutions and this leads to multiple many solutions (at least three). We discuss the numerical results obtained for different values of the exponents pj and the applied voltage on different meshes.

Adaptive Concurrent Learning Algorithm Based on Pontryagin's Maximum Principle for Nonlinear System Optimal Tracking Control with State Inequality Constraints

Adaptive Concurrent Learning Algorithm Based on Pontryagin's Maximum Principle for Nonlinear System Optimal Tracking Control with State Inequality Constraints