Maximum Principle Research Articles

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

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

Performance analysis of agricultural machinery with aid digital agriculture

Digital agriculture has increasingly gained strength in the countryside, as it is considered a byproduct of growing globalization. As the search for production optimization, cost reduction, and productivity increase continues, these tools become the great allies of the producer in the final agricultural production process.Therefore, agricultural operations must also follow this evolution in the development of technological improvements, which carry positive aspects for large-scale production. In this context, the objective of this work was to evaluate the performance of seeders-fertilizers with different input distribution mechanisms. Since one of the biggest expenses in the production process is the acquisition of seeds and fertilizer, any savings in production costs become a significant advantage for the producer, who will have a greater net income. The experimental design used to evaluate the data followed the premises of Statistical Process Control (SPC), and the results were assessed using individual value control charts. It was concluded that the production process was negatively affected by the performance of the seeders-fertilizers, as in one area, more seeds were falling than necessary, increasing the cost per hectare and not adhering to the principles of cost minimization and profit maximization. Conversely, in the second area evaluated, the machinery's performance resulted in fewer seeds being deposited than planned. Consequently, according to the results, the producer lost the opportunity to produce 5.85 sc/ha without reducing costs in the operation, as the seeder simply did not deposit the seeds, thereby reducing the gain per area, which could have been more productive, increasing final profitability.

Burst pressure prediction of cord-rubber composite structures using global–local nonlinear finite element analysis

This study aims to develop a model to predict the burst pressure of a dry filament wound cord-rubber composite pressure vessel under hydrostatic internal pressurization using a submodelling based global–local FEA model. The model links the global displacements of a rebar-based model to obtain the local deformation state in a single rhomboidal representative volume. Emphasis is placed on capturing the local stress concentrations in the fibers due to the unique filament winding mosaic pattern. Fiber damage is included in the local model using a maximum principle strain criteria. Verification of the created model is done experimentally on industrially manufactured burst-test specimens. Measurements for displacement during the experiments are taken photographically, while the burst pressure is captured using a pressure transducer. The final error between the burst pressure of the samples and the experimental demonstrators is approximately 6.5%, a marked improvement over conventional models with truss and rebar elements as fibers.

New limiter regions for multidimensional flows

Accurate transport algorithms are crucial for computational fluid dynamics and more accurate and efficient schemes are always in development. One dimensional limiting is commonly employed to suppress nonphysical oscillations. However, the application of such limiters can reduce accuracy. It is important to identify the weakest set of sufficient conditions required on the limiter as to allow the development of successful numerical algorithms.The main goal of this paper is to identify new less restrictive sufficient conditions for flux form in-compressible advection to remain monotonic. We identify additional necessary conditions for incompressible flux form advection to be monotonic, demonstrating that the Spekreijse limiter region is not sufficient for incompressible flux form advection to remain monotonic. Then a convex combination argument is used to derive new sufficient conditions that are less restrictive than the Sweby region for a discrete maximum principle. This allows the introduction of two new more general limiter regions suitable for flux form incompressible advection.

Comprehensive Evaluation of Crack Safety of Hydraulic Concrete Based on Improved Combination Weighted-Extension Cloud Theory

When multiple elements come together, hydraulic concrete develops cracks of varying widths, which huts the dependability of buildings. Therefore, with pertinent tools or procedures, swiftly ascertaining the safety status of hydraulic concrete cracks under diverse service conditions is required by conducting a quantitative and qualitative analysis of the elements influencing the onset of cracks. This paper took the safety status of hydraulic concrete cracks as the main body of research; every step of hydraulic conservation infrastructure from the ground up—design stage, construction process, operation environment, and impoundment operation—was thoroughly examined. After establishing a multi-dimensional and multi-level system for the safety status evaluation of hydraulic concrete cracks, the subjective exponential AHP and objective CRITIC method were employed to determine the weight of each factor. Then, the two weights were processed using an enhanced combination assignment method to produce a more scientifically developed combination weight. Furthermore, fuzziness and randomness were considered in the quantitative analysis thanks to integrating cloud theory and extension matter elements. In order to determine the safety evaluation findings for hydraulic concrete fractures, the maximum membership principle and the cloud picture were employed. The conclusion reached after using this method to evaluate Dianzhan Dam was that the crack had a safety grade of III, meaning that it greatly impacted the reliability of the dam, and called for prompt acceptance or repair measures to improve building efficiency and safety.

Optimal Control Analysis in the Treatment of Solid Tumors using Combined Therapy

In this paper, we develop and analyze a mathematical model that describes the dynamic interactions and competitions among tumor cells, normal cells, immune cells, and transforming growth factor-beta within the tumor microenvironment. We conducted qualitative analyses to examine the persistence or extinction of each cell population and analyzed the regions of stability and instability across various equilibria. Additionally, we formulated and solved an optimal control problem using the Pontryagin’s maximum principle, aiming to minimize tumor size and the concentration of transforming growth factor-beta while also reducing chemotherapy and siRNA drug-induced toxicity in patients. Numerical simulations are performed for the model with and without treatment. We demonstrate scenarios where neither individual treatment is capable of reducing both tumor and TGF-β, but their combination achieves a substantial reduction.

Inferring potential landscapes: A Schrödinger bridge approach to maximum caliber

Schrödinger bridges have emerged as an enabling framework for unveiling the stochastic dynamics of systems based on marginal observations at different points in time. The terminology “bridge” refers to a probability law that suitably interpolates such marginals. The theory plays a pivotal role in a variety of contemporary developments in machine learning, stochastic control, thermodynamics, and biology, to name a few, impacting disciplines such as single-cell genomics, meteorology, and robotics. In this work, we extend Schrödinger's paradigm of bridges to account for integral constraints along paths, in a way akin to maximum caliber—a maximum entropy principle applied in a dynamic context. The maximum caliber principle has proven useful to infer the dynamics of complex systems, e.g., model gene circuits and protein folding. We unify these two problems via a maximum likelihood formulation to reconcile stochastic dynamics with ensemble-path data. A variety of data types can be encompassed, ranging from distribution moments to average currents along paths. The framework enables inference of time-varying potential landscapes that drive the process. The resulting forces can be interpreted as the optimal control that drives the system in a way that abides by specified integral constraints. This, in turn, relates to a similarly constrained optimal mass transport problem in the zero-noise limit. Analogous results are presented in a discrete-time, discrete-space setting and specialized to steady-state dynamics. We finish by illustrating the practical applicability of the framework through paradigmatic examples, such as that of bit erasure or protein folding. In doing so, we highlight the strengths of the proposed framework, namely, the generality of the theory, its elegant analytical structure, the ease of computation, and the ability to interpret results in terms of system dynamics. This is in contrast to maximum caliber problems where the focus is typically on updating a probability law on paths. Published by the American Physical Society 2024

Mathematical modeling, sensitivity analysis, and optimal control of students awareness in mathematics education

This article presents a continuous-time mathematical model, EFSMK, integrating various educational categories, serving as a valuable tool for understanding students interactions with mathematics. The paper begins by exploring the necessary preliminaries to describe the model. Subsequently, we examine the model’s well-posedness, focusing on the positivity and boundedness of solutions. The basic reproduction number is then derived using the next-generation matrix method. The model exhibits two stable states: the mathematics-free equilibrium point and the learning equilibrium point for students who struggle with mathematics. The study shows that when the basic reproduction number is less than one, the mathematics-free equilibrium point is globally asymptotically stable. Conversely, when the basic reproduction number exceeds one, the learning equilibrium point for students who struggle with mathematics is globally asymptotically stable. Numerical simulations validate the theoretical findings. In the last part of the paper, we present a structured model involving two distinct strategies aimed at improving the mathematical skills of students who encounter difficulties in this subject. The first strategy consists of offering individualized or small-group tutoring sessions, focusing on challenging topics and using alternative teaching methods. The second strategy entails adjusting the overall program and pedagogical approach to better respond to diverse learning needs. This approach uses Pontryagin’s maximum principle and iterative analysis to determine optimal controls. The effectiveness of these strategies is evaluated using numerical simulations in various scenarios.

Exponential stability and optimal control of a stochastic brucellosis model with spatial diffusion and nonlocal transmission

In this paper, a stochastic brucellosis model with nonlocal transmission and spatial diffusion is established. Existence, uniqueness and positivity of mild solution to the model are obtained by adopting a truncation method. To ensure the Mean Square Exponential Stability (MSES) of the solution, sufficient conditions are given by employing the inequality techniques and the stability implies that the brucellosis die out in a short period of time. Moreover, we introduce elimination of infected animals and disinfection of brucella to the model as control strategies. Necessary conditions are given to obtain the optimal control which best balance the outcomes and costs of the control by applying Pontryagin’s maximum principle. Finally, the theoretical results are demonstrated by the numerical simulations.

Correction: Variational maximum principle for elliptic systems involving the fractional Laplacian

Correction: Variational maximum principle for elliptic systems involving the fractional Laplacian

Modeling the transmission dynamics of the co-infection of COVID-19 and Monkeypox diseases with optimal control strategies and cost–benefit analysis

This study explores a mathematical model in continuous time, elucidating the transmission dynamics of the co-infection of COVID-19 and Monkeypox. We implemented an optimal control strategy that includes adherence to wearing nose/face masks, practicing social distancing, employing rodenticides, and getting vaccinated against these lethal illnesses. The goal is to reduce their spread among people, consequently lessening their impact on the human population. Utilizing the discrete-time Pontryagin maximum principle, we determined optimal control strategies through an iterative approach to solve the optimal system. By utilizing the MATLAB optimization toolbox with the Runge–Kutta forward–backward sweep technique, we performed numerical simulations to demonstrate how these control parameters impact different sections of the human population, including both infected and uninfected compartments. We conducted a comprehensive cost–benefit analysis to identify the control measures that would produce the best outcome in minimizing both the expenses and the incidence of co-infection cases involving COVID-19 and Monkeypox diseases. Our analyses indicated that intensifying the utilization of nose/facemasks, rodenticides, personal protective equipments (PPEs) in Monkeypox treatment facilities, immunization, and implementing social distancing measures can notably diminish the prevalence of COVID-19, Monkeypox, and their concurrent infections among the human populace.

The applicability of Zermelo's equation to indirect and direct optimization of commercial aircraft flights

This paper focuses on the applicability of Zermelo's equation within the context of 3D commercial aircraft trajectory optimization problems. The associated optimal control problem includes two singular controls, specifically aerodynamic path angle and throttle setting, and one regular control, specifically heading angle. Using Pontryagin's maximum principle and the direct adjoining method, we show that the optimal heading angle is defined by Zermelo's equation. The significance of the presented analysis is that Zermelo's equation holds for 3D commercial aircraft flights, even in the presence of standard state-inequality constraints and a more general objective function. With the help of Zermelo's equation, the control problem is initially analyzed for a 3D time-optimal climb-phase scenario, which is solved by an indirect approach. Through the analysis of the switching function, we demonstrate the dependency of the optimal aerodynamic path angle on the optimal heading angle. Next, by tackling a 3D time-fuel-optimal free-routing flight, solved by a direct approach, we show that Zermelo's equation can help reduce the overall dimension of the associated nonlinear programming. To successfully handle the initial guess in both indirect and direct problems, it is proposed a diminutive nonlinear programming technique as a fast and robust initializer. This simple-yet-effective initializer provides sufficient information about the optimal controls and state dynamics required for initializing a direct optimization, as well as the optimal switching times and co-states required for initializing an indirect optimization.

Discovering Maximum Entropy Knowledge

AbstractThis paper seeks to determine a rational agent’s evidential constraints given her beliefs. Rationality is here construed as adherence to a principle of entropy maximisation. I determine the rational agent’s set of probability efunctions compatible with the evidence, $${\mathbb {E}}$$ E , given the maximum entropy function and given some constraints on the shape of $${\mathbb {E}}$$ E . I also consider agents employing a centre of mass approach to form their beliefs rather than entropy maximisation.

Maximum principle preserving time implicit DGSEM for linear scalar hyperbolic conservation laws

The properties of the high-order discontinuous Galerkin spectral element method (DGSEM) with implicit backward Euler time stepping are investigated for the approximation of hyperbolic linear scalar conservation equation in multiple space dimensions. We first prove that the DGSEM scheme in one space dimension preserves a maximum principle for the cell-averaged solution when the time step is large enough. This property however no longer holds in multiple space dimensions and we propose to use the flux-corrected transport (FCT) limiting [5] based on a low-order approximation using graph viscosity to impose a maximum principle on the cell-averaged solution. These results allow us to use a linear scaling limiter [58] in order to impose a maximum principle at nodal values within elements, while limiting the cell average with the FCT limiter improves the accuracy of the limited solution. Then, we investigate the inversion of the linear systems resulting from the time implicit discretization at each time step. We prove that the diagonal blocks are invertible and provide efficient algorithms for their inversion. Numerical experiments in one and two space dimensions are presented to illustrate the conclusions of the present analyses.

Modeling and optimal control of COVID-19 with comorbidity and three-dose vaccination in Indonesia

This paper presents and examines a COVID-19 model that takes comorbidities and up to three vaccine doses into account. We analyze the stability of the equilibria, examine herd immunity, and conduct a sensitivity analysis validated by data on COVID-19 in Indonesia. The disease-free equilibrium is locally and globally asymptotically stable whenever the basic reproduction number is less than one, while an endemic equilibrium exists and is globally asymptotically stable when the number is greater than one. Subsequently, the model incorporates two effective measures, namely public education and enhanced medical care, to determine the most advantageous approach for mitigating the transmission of the disease. The optimal control model is then determined using Pontryagin’s maximum principle. The integrated control strategy is the best method for reliably safeguarding the general population against COVID-19 infection. Cost evaluations and numerical simulations corroborate this conclusion.

An optimal control vaccine model of COVID-19 with cost-effective analysis

This study presents a comprehensive COVID-19 vaccination model, integrating breakthrough transmission, and evaluates its efficacy through various analytical techniques. To assess secondary infections, four effective reproduction rates are computed corresponding to different disease states. The sensitivity analysis of the reproduction rates is conducted using the PRCC to identify influential compartments and parameters. The model is extended to incorporate five time-dependent control interventions to reduce infected compartments, with optimisation performed using the Pontryagin maximum principle. A cost-effectiveness analysis of these strategies is conducted using the average cost-effectiveness ratio. Analysis shows that the model does not undergo backward bifurcation, meaning a stable disease-free equilibrium cannot coexist with a stable endemic equilibrium. The findings show that strict adherence to standard operating procedures is the most effective single intervention. In contrast, a combined approach incorporating all five control interventions is the most effective in averting infections with the lowest ACER.

Optimal control of rotavirus infection in breastfed and non-breastfed children

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.

Analyzing the use of non-pharmaceutical personal protective measures through self-interest and social optimum for the control of an emerging disease

Non-pharmaceutical personal protective (NPP) measures such as face masks use, and hand and respiratory hygiene can be effective measures for mitigating the spread of aerosol/airborne diseases, such as COVID-19, in the absence of vaccination or treatment. However, the usage of such measures is constrained by their inherent perceived cost and effectiveness for reducing transmission risk. To understand the complex interaction of disease dynamics and individuals decision whether to adopt NPP or not, we incorporate evolutionary game theory into an epidemic model such as COVID-19. To compare how self-interested NPP use differs from social optimum, we also investigated optional control from a central planner’s perspective. We use Pontryagin’s maximum principle to identify the population-level NPP uptake that minimizes disease incidence by incurring the minimum costs. The evolutionary behavior model shows that NPP uptake increases at lower perceived costs of NPP, higher transmission risk, shorter duration of NPP use, higher effectiveness of NPP, and shorter duration of disease-induced immunity. Though social optimum NPP usage is generally more effective in reducing disease incidence than self-interested usage, our analysis identifies conditions under which both strategies get closer. Our model provides new insights for public health in mitigating a disease outbreak through NPP.

Two-person zero-sum game with terminal state constraints

In this paper, the discrete-time two-person zero-sum game with terminal state constraints is considered. By applying the variational method, the maximum principle for zero-sum game with terminal constraint is derived along with the forward and backward difference equations (FBDEs). The main contributions of this paper are two-fold. On the one hand, by decoupled solving the FBDEs, the analytical solution to the zero-sum game with terminal state constraint is given. On the other hand, the necessary and sufficient conditions for the reachability of the system are obtained. In this way, the solvability of the zero-sum game with terminal state constraint is re-expressed by the reachability of the system.

Existence of Viscosity Solutions for Weakly Coupled Cooperative Parabolic Systems with Fully Nonlinear Principle Part

In this paper, the existence of viscosity solutions for weakly coupled, degenerate, and cooperative parabolic systems is studied in a bounded domain. In particular, we consider the viscosity solutions of parabolic systems with fully nonlinear degenerated principal symbol and linear coupling part. The maximum principle theorem is given as well.