Maximum Principle Research Articles

Dynamic reliability analysis of angular contact ball bearing based on maximum entropy quadratic fourth moment method*

ABSTRACT Bearing failure occurs when the dynamic contact stress exceeds the yield strength of the bearing material during operation. To minimise the risk of bearing failure, a new reliability theory method is proposed based on the stress-strength model of bearings, incorporating random input variables to analyse the reliability of angular contact ball bearings (ACBBs) and quantify their dynamic performance. The maximum entropy quadratic fourth moment method is used to calculate the probability distribution of maximum orthogonal shear stress-strength of bearing, and the maximum entropy principle is used to obtain the approximation of the minimum human error factors. Additionally, reliability-based sensitivity indices are derived to investigate the parametric significance of random input variables. Using B218 bearing as an example, the engineering application of this method in dynamic reliability and reliability-based sensitivity analysis is illustrated. Monte Carlo simulation is performed to verify the accuracy and effectiveness of the method to provide benchmark results. The results show that the Poisson’s ratio have a greater influence on the dynamic reliability of ACBBs than other parameters. Improving parameters such as ball diameter, raceway groove curvature radius and contact angle can also improve the reliability of bearings to some extent.

Maximum principles for elliptic operators in unbounded Riemannian domains

The necessity of a Maximum Principle arises naturally when one is interested in the study of qualitative properties of solutions to partial differential equations. In general, to ensure the validity of these kinds of principles one has to consider some additional assumptions on the ambient manifold or on the differential operator. The present work aims to address, using both of these approaches, the problem of proving Maximum Principles for second order, elliptic operators acting on unbounded Riemannian domains under Dirichlet boundary conditions. Hence there is a natural division of this article in two distinct and standalone sections.

On the maximum principle and high-order, delay-free integrators for the viscous Cahn–Hilliard equation

On the maximum principle and high-order, delay-free integrators for the viscous Cahn–Hilliard equation

Reliable optimal controls for SEIR models in epidemiology

We present and compare two different optimal control approaches applied to SEIR models in epidemiology, which allow us to obtain some policies for controlling the spread of an epidemic. The first approach uses Dynamic Programming to characterise the value function of the problem as the solution of a partial differential equation, the Hamilton–Jacobi–Bellman equation, and derive the optimal policy in feedback form. The second is based on Pontryagin’s maximum principle and directly gives open-loop controls, via the solution of an optimality system of ordinary differential equations. This method, however, may not converge to the optimal solution. We propose a combination of the two methods in order to obtain high-quality and reliable solutions. Several simulations are presented and discussed, also checking first and second order necessary optimality conditions for the corresponding numerical solutions.

Optimal control problem with state constraints via penalty functions

In this paper, we derive necessary conditions of optimality for problems with state constraints using the method of penalty functions similar to the one we previously used to solve optimization problems for control sweeping processes (see, e.g., de Pinho et al., 2022). Our aim is to provide a proof of the Maximum Principle that can be easily followed by those with basic knowledge of classical optimal control and elementary functional analysis. We intentionally consider a smooth case and the simplest boundary conditions; we consider global minimum and assume that the set of trajectories of the control system is compact.

Introduction to Theoretical and Experimental aspects of Quantum Optimal Control

Abstract Quantum optimal control is a set of methods for designing time-varying electromagnetic fields to perform operations in quantum technologies. This tutorial paper introduces the basic elements of this theory based on the Pontryagin maximum principle, in a physicist-friendly way. An analogy with classical Lagrangian and Hamiltonian mechanics is proposed to present the main results used in this field. Emphasis is placed on the different numerical algorithms to solve a quantum optimal control problem. Several examples ranging from the control of two-level quantum systems to that of Bose-Einstein Condensates (BEC) in a one-dimensional optical lattice are studied in detail, using both analytical and numerical methods. Codes based on shooting method and gradient-based algorithms are provided. The connection between optimal processes and the quantum speed limit is also discussed in two-level quantum systems. In the case of BEC, the experimental implementation of optimal control protocols is described, both for two-level and many-level cases, with the current constraints and limitations of such platforms. This presentation is illustrated by the corresponding experimental results.

Mathematical modeling and optimal control strategies to limit the spread of fowl pox in poultry

This paper presents a mathematical model aimed at elucidating the dynamics of fowl pox transmission within poultry populations. The model focuses on implementing control strategies to manage the spread of the disease, considering two primary modes of transmission: direct contact and transmission via mosquitoes. Our objective is to enhance understanding of disease propagation and to propose a control strategy aimed at minimizing the number of infected birds Ib(t) and increasing the number of recovered birds Rb(t) during the time interval t0,tf, while also minimizing the costs associated with implementing the control strategy. The proposed mathematical framework facilitates the integration of various control strategies to effectively manage fowl pox transmission dynamics. By employing Pontryagin’s maximum principle, efficient control measures are identified to mitigate the spread of the disease. Finally, numerical simulations are performed to verify the theoretical analysis using MATLAB.

An approach for end-to-end optimization of low-thrust interplanetary trajectories using collinear libration points

This article presents an approach to solve the problem of end-to-end optimization of the low-thrust interplanetary trajectory. At first, the problem of optimizing low-thrust interplanetary trajectory, which passes through the collinear libration points near the planets of departure and arrival is considered. The obtained solutions from this problem can be used as an initial guess to solve the end-to-end optimization problem, i.e., to calculate interplanetary low-thrust trajectories that satisfy the necessary optimality conditions at the match points of the heliocentric and planetocentric segments of trajectory. The minimum-fuel problem is considered. An indirect method based on the maximum principle, and the continuation method is applied to optimize interplanetary spacecraft trajectories. Numerical examples of trajectories from the Earth orbit to the orbit around the destination planet (Mars and Jupiter) are given. The possibility of a significant reduction in the required characteristic velocity is shown in comparison with the estimates obtained by using the zero-radius sphere of influence model.

Sustainable management of predatory fish affected by an Allee effect through marine protected areas and taxation

Ecological balance and stable economic development are crucial for the fishery. This study proposes a predator–prey system for marine communities, where the growth of predators follows the Allee effect and takes into account the rapid fluctuations in resource prices caused by supply and demand. The system predicts the existence of catastrophic equilibrium, which may lead to the extinction of prey, consequently leading to the extinction of predators, but fishing efforts remain high. Marine protected areas are established near fishing areas to avoid such situations. Fish migrate rapidly between these two areas and are only harvested in the nonprotected areas. A three-dimensional simplified model is derived by applying variable aggregation to describe the variation of global variables on a slow time scale. To seek conditions to avoid species extinction and maintain sustainable fishing activities, the existence of positive equilibrium points and their local stability are explored based on the simplified model. Moreover, the long-term impact of establishing marine protected areas and levying taxes based on unit catch on fishery dynamics is studied, and the optimal tax policy is obtained by applying Pontryagin’s maximum principle. The theoretical analysis and numerical examples of this study demonstrate the comprehensive effectiveness of increasing the proportion of marine protected areas and controlling taxes on the sustainable development of fishery.

A nonlinear finite volume element method preserving the discrete maximum principle for heterogeneous anisotropic diffusion equations

In this paper, we propose a new nonlinear conservative and maximum-principle-preserving finite volume element method for heterogeneous anisotropic diffusion equations on triangular or strictly convex quadrilateral meshes. First, we decompose the numerical flux of the standard finite volume element method on each dual edge into two parts: main part having a two-point-flux structure, and residual part owning a multi-point-flux structure. Then, according to the sign of the residual part and the distribution of the local extremum, the residual part is modified to a two-point-flux structure with a nonlinear term. The resulting nonlinear scheme not only inherits conservation and accuracy of original scheme, but also possesses a maximum-preserving structure without extra restrictions on the meshes and diffusion coefficients. Several numerical experiments verify that our nonlinear scheme has an optimal convergence order and satisfies the discrete maximum principle.

Optimal switching for Networked Control Systems with information multiplexing

In this article, we examine a Networked Control System (NCS) in which the plant and the controller communicate over a network subject to a certain communication constraint. The plant is described by discrete-time nonlinear dynamics subject to bounded disturbances. Due to an overloaded communication network, we assume that the control signal and the information from the plant (the measured output signal) cannot be transmitted simultaneously and are subject to a multiplexing constraint. The goal is to design a switching strategy that allows us to sequentially communicate given these constraints while optimizing a quadratic cost over a finite horizon. Consequently, we proceed by emulation and assume that a controller that satisfies performance requirements is already provided. The resulting optimization problem is observed to be an integer programming problem that is generally NP-complete, i.e., the complexity is exponential in the time horizon considered. To overcome this issue, we provide a different perspective on this problem than what has been presented by the community before. Our main contribution is to reformulate the problem with all its constraints to a form that renders it amenable to apply the discrete-time Pontryagin Maximum Principle to get the necessary conditions for the optimality of the control action sequence. These necessary conditions are then solved numerically by a multiple-shooting method. To validate the approach, we present some illustrative numerical experiments on an inverted pendulum. Different setups are considered and numerically analyzed: usage of a predictor when the output is not transmitted and usage of the previous value of the output when the new value is not transmitted, with or without the choice of non-transmission.

Pareto efficiency of infinite-horizon cooperative stochastic differential games with Markov jumps and Poisson jumps

This paper investigates a wide class of infinite-horizon cooperative stochastic differential games with Markov jumps and Poisson jumps, which can model uncertainties of internal mode transition and characterize occasional sudden changes in the games, respectively. Firstly, the necessary conditions which guarantee the existence of Pareto solutions are obtained by utilizing the Lagrange multiplier method and the stochastic maximum principle with Markov jumps and Poisson jumps. Then the sufficient conditions which guarantee the existence of Pareto efficient strategies are derived. Secondly, the well-posedness of cooperative stochastic differential games (CSDG) with Markov jumps and Poisson jumps in infinite horizon is established when the solution of generalized algebraic Riccati equation (GARE) exists. Furthermore, we can obtain Pareto solutions by introducing the coupled algebraic Lyapunov equations (ALEs). Finally, the numerical examples verify the theoretical results.

Stochastic degradation modeling and remaining useful lifetime prediction based on long short-term memory network

In this paper, a novel remaining useful lifetime (RUL) prediction method that fuses stochastic degradation modeling and machine learning is proposed to improve the fitness of the model and quantify the uncertainty of the prediction results. First, a stochastic degradation model based on the Wiener process is built, and the drift increment is extracted using empirical mode decomposition (EMD). Second, a long short-term memory (LSTM) network is trained to learn the equipment degradation rule and predict the drift increment. The diffusion coefficient of the degradation model is then estimated according to the maximum likelihood principle. The final step is to derive the analytical expression for the probability distribution of remaining useful lifetime (RUL) based on the concept of first hitting time and the difference principle. The lithium battery degradation test confirmed the efficacy of the proposed method, achieving a life cycle average prediction accuracy of up to 97.45%.

A Deformation Analysis Method for Sluice Structure Based on Panel Data

Deformation, as the most intuitive index, can reflect the operation status of hydraulic structures comprehensively, and reasonable analysis of deformation behavior has important guiding significance for structural long-term service. Currently, the health evaluation of dam deformation behavior has attracted widespread attention and extensive research from scholars due to its great importance. However, given that the sluice is a low-head hydraulic structure, the consequences of its failure are easily overlooked without sufficient attention. While the influencing factors of the sluice’s deformation are almost identical to those of a concrete dam, nonuniform deformation is the key issue in the sluice’s case because of the uneven property of the external load and soil foundation, and referencing the traditional deformation statistical model of a concrete dam cannot directly represent the nonuniform deformation behavior of a sluice. In this paper, we assume that the deformation at various positions of the sluice consist of both overall and individual effects, where overall effect values describe the deformation response trend of the sluice structure under external loads, and individual effect values represent the degree to which the deformation of a single point deviates from the overall deformation. Then, the random coefficient model of panel data is introduced into the analysis of sluice deformation to handle the unobservable overall and individual effects. Furthermore, the maximum entropy principle is applied, both to approximate the probability distribution function of individual effect extreme values and to determine the early warning indicators, completing the assessment and analysis of the nonuniform deformation state. Finally, taking a project as an example, we show that the method proposed can effectively identify the overall deformation trend of the sluice and the deviation degree of each measuring point from the overall deformation, which provides a novel approach for sluice deformation behavior research.

Convex hull property for elliptic and parabolic systems of PDE

We study the convex hull property for systems of partial differential equations. This is a generalization of the maximum principle for a single equation. We show that the convex hull property holds for a class of elliptic and parabolic systems of non-linear partial differential equations. In particular, this includes the case of the parabolic p-Laplace system. The coupling conditions for coefficients are demonstrated to be optimal by means of respective counterexamples.

Periodic on-off operations of the self-cycling ethanol fermentation process: Stability analysis and optimal operation strategies

Fuel ethanol is highlighted for being renewable and carbon neutral, but a multi-scale challenge exists that hampers its role as a future energy substitute, and with the breakthrough of cellulose pretreatment and enzyme hydrolysis technology, fermentation itself has become one of the main limiting factors for process enhancement. To tackle the problem, this paper studies periodic operation strategies to improve process performance and avoid possible instabilities. Firstly, superiority by periodic operation over the corresponding steady-state operation is verified. Then, to settle the problem of possible instabilities under impulsive control, stability of state-impulsive control is conducted and for the studied case, any self-cycling fermentation starts from batch mode with 50 % discharge-and-refill portion exhibits superior repeatability property, which allows continuous fermentation to operate under very slow overall dilution rate to cope with the extreme slow fermentation rate. Finally, on-off operation problem is formulated that is governed by a multidimensional nonlinear system, affine to the manipulated variable that is under the isoperimetric constraint; with the use of Pontryagin maximum principle, the optimal synthesis of the optimal strategies under integral constraint is computed out, and a practically plausible on-off operation strategy for ethanol fermentation is formulated with guaranteed periodic stability and optimality.

Global attractivity for reaction–diffusion equations with periodic coefficients and time delays

In this paper, we provide sharp criteria of global attraction for a class of non-autonomous reaction–diffusion equations with delay and Neumann conditions. Our methodology is based on a subtle combination of some dynamical system tools and the maximum principle for parabolic equations. It is worth mentioning that our results are achieved under very weak and verifiable conditions. We apply our results to a wide variety of classical models, including the non-autonomous variants of Nicholson’s equation or the Mackey–Glass model. In some cases, our technique gives the optimal conditions for the global attraction.

Energy stable and maximum bound principle preserving schemes for the Allen-Cahn equation based on the Saul’yev methods

Energy stable and maximum bound principle preserving schemes for the Allen-Cahn equation based on the Saul’yev methods

Phase stability and mechanical properties of the six-principal element TiVNbCrCoNi alloys

This study designed a series of six-principal element TiVNbCrCoNi alloys according to the maximum entropy principle under the constraint of valence electron concentration (VEC). The phase stability and mechanical properties were investigated. It was showed that at VEC levels below 5.2, the body-centered cubic (BCC)-structured solid-solution phase remained stable at room temperature. As VEC exceeded 8.0, the face-centered cubic (FCC)-structured solid-solution phase stabilized. In the VEC range of 5.2–8.0, (Co, Ni)Ti and Ni3Nb intermetallic compounds formed combining with the solid-solution phases. Both yield strength and ductility increased with increase of VEC in the BCC-structured solid-solution alloys. The phase stability was elucidated through First-principle and thermodynamic calculations, while the evolution of mechanical properties of solid-solution alloys was evaluated in terms of multiple parameters. The thermodynamic calculation achieved satisfactory accuracy after being adjusted to utilize atomic bond enthalpy instead of atomic electronegativity difference.

Dynamics of a discrete one‐predator two‐prey system with Michaelis–Menten‐type prey harvesting and prey refuge

In this paper, we propose and study a discretized one‐predator two‐prey system along with prey refuge and Michaelis–Menten‐type prey harvesting. The interaction among the species is considered as Holling type III functional response. Firstly, existence and local stability of all the fixed points are derived under certain parametric conditions. Furthermore, a special consideration is made to global asymptotic stability of the interior fixed point. Then, we have shown that the system undergoes different types of bifurcations including transcritical bifurcation, flip bifurcation, and Neimark–Sacker bifurcation by using center manifold theorem, bifurcation theory, and normal form method. Also, Feigenbaum's constant of the system is calculated. It is observed that both harvesting and refuge have a stabilizing effect on the system, and the stabilizing effect of harvesting dominates the stabilizing effect of refuge. Of most interest is the finding of coexisting attractors and multistability. In particular, optimal harvesting policy has been obtained by extension of Pontryagin's maximum principle to discrete system. Finally, some intriguing numerical simulations are provided to verify our analytic findings and rich dynamics of the three species system.