Pontryagin's Principle Research Articles

Stability Analysis and Optimal Control for Spreading Typhoid Fever Model with Direct and Indirect Transmissions

This article discusses a model of spreading typhoid fever with direct and indirect transmissions. There are five compartments included in the model, namely susceptible individuals, infected individuals, chronic carrier individuals, recovered individuals, and salmonella typhi bacteria in the environment. As an effort to control and minimize the numbers of infected individuals and chronic carrier individuals, we introduce simultaneously campaign for the susceptible individuals and treatment for the infected individuals and for chronic carrier individuals. The resulting model is a system of non-linear differential equations and quite challenging to analysis globally. Existence of both disease-free equilibrium point and endemic equilibrium point are analysed analytically. Stability of the equilibrium point is analysed locally by determining the eigenvalues of the associated Jacobian matrix, Routh-Hurwitz stability criteria, and basic reproduction number () via next generation matrix. The minimum Pontryagin principle and Hamiltonian equation are referred to minimize the numbers of infected and chronic carrier individuals. Simulation is carried out using suitable values of parameters. We found that the eigenvalues for endemic equilibrium point are all negative real numbers and . Optimal paths for state and constate variables are plotted using fourth order forward-backward Runge-Kutta method. In case there are no campaign and treatments, we found and endemic occurs. When campaign and treatments are included together in the model, we found optimal paths for all compartments that minimize the number of infected and chronic carrier individuals. Giving campaign and treatments increase significantly the susceptible and recovered individuals. At the same time, it reduces significantly the infected, chronic carrier individuals, and salmonella typhi bacteria in the environment. Involving campaign and treatment in the model of spreading typhoid fever can be considered as an effective strategy to minimize the numbers of infected and chronic carrier individuals.

Isoperimetric Constraint Inference for Discrete-Time Nonlinear Systems Based on Inverse Optimal Control.

In this article, the problem of inferring unknown isoperimetric constraints is considered given optimal state and control trajectories that solve the optimal control problem with isoperimetric constraints. By exploiting Pontryagin's principle, the recovery equations for unknown isoperimetric constraints are established. Under verifiable dimensionality condition and matrix rank condition, the proposed method is guaranteed to infer the unknown isoperimetric constraints exactly. Furthermore, the proposed method is extended to multiple trajectory setting. Finally, the effectiveness of the proposed method is illustrated by two simulation examples with various settings.

Nonlinear SIRS Fractional-Order Model: Analysing the Impact of Public Attitudes towards Vaccination, Government Actions, and Social Behavior on Disease Spread

This present work develops a nonlinear SIRS fractional-order model with a system of four equations in the Caputo sense. This study examines the impact of positive and negative attitudes towards vaccination, as well as the role of government actions, social behavior and public reaction on the spread of infectious diseases. The local stability of the equilibrium points is analyzed. Sensitivity analysis is conducted to calculate and discuss the sensitivity index of various parameters. It has been established that the illness would spread across this system when the basic reproduction number is larger than 1, the system becomes infection-free when the reproduction number lies below its threshold value of 1. Numerical figures depict the effects of positive and negative attitudes towards vaccination to make the system disease-free sooner. A comprehensive study regarding various values of the order of fractional derivatives together with integer-order derivatives has been discussed in the numerical section to obtain some useful insights into the intricate dynamics of the proposed system. The Pontryagin principle is used in the formulation and subsequent discussion of an optimum control issue. The study also reveals the significant role of government actions in controlling the epidemic. A numerical analysis has been conducted to compare the system’s behavior under optimal control and without optimal control, aiming to discern their differences. The policies implemented by the government are regarded as the most adequate control strategy, and it is determined that the execution of control mechanisms considerably diminishes the ailment burden.

Применение принципа Понтрягина для оптимального отключения энергетического ядерного реактора

The theory of optimal control is based on two approaches, the dynamic programming method (Bellman equation) and Pontryagin maximum principle. Pontryagin maximum principle is applied in the physics of nuclear reactors when optimizing various transient processes. The mathematical justification of this theory is based on the elements of convex analysis, which is not used by physicists and engineers, so the physical issues are less studied in scientific literature. The subject of the study is a nuclear power reactor of the WWER type. The problem of minimizing its shutdown time, bypassing the iodine pit is solved and it is possible to start up the reactor at any moment after its shutdown. Analytical and numerical methods are used. The paper considers an example of applying the maximum principle for optimal control of the process of shutting down a nuclear reactor bypassing the iodine pit. A physical mathematical model of the Pontryagin principle is formulated. The process of optimal control of reactor shutdown for large and small reactivity margins is justified and calculated. Pontryagin principle does not contain an algorithm to find an optimization process; the stages of the process must be selected based on physical considerations, but these stages must satisfy the specified principle. Based on the Pontryagin principle, the results of the study make it possible to draw up a step-by-step action plan when shutting down a WWER-type reactor with any value of the reactivity margin and its switching is possible at any time after the transition process, which avoids downtime. The proposed plan can be used both in mathematical modeling of transient processes in a reactor and in reactor control systems to improve its controllability and, consequently, to improve safety.

Comparison of Optimal Control Effect from Fungicides and Pseudomonas Fluorescens on Downy Mildew in Corn

Downy mildew is a disease that continues to infect corn crops in Timor Tengah Utara regency, reducing the amount of crop production and making corn farmers suffer losses. Farmers continue to look for ways to control downy mildew. Two treatments are commonly used by farmers, namely spraying Fungicides and Pseudomonas Fluorescens simultaneously in one unit of time, but have not resulted in optimal production. Therefore, this research is important to get a more optimal way to control find downy mildew. In this paper, we determine the optimal model of downy mildew control in corn plants by comparing the use of Fungicides and Pseudomonas Fluorescens. This research begins by forming a dynamic mathematical model consisting of six populations, namely four corn populations (Sh, F , P , Ih) and two populations of infecting fungi (Sv , Iv ). Then they obtained the basic reproduction number (R0) and two equilibrium points, namely the disease-free equilibrium and the disease-endemic equilibrium which has asymptotic stability. Numerical simulation results based on optimal control analysis with the minimum Pontryagin principle show that using fungicides can reduce the number of plants infected with downy mildew. Therefore, control by using fungicides is necessary and recommended increasing the number of downy mildew infected plants.

RUNGE–KUTTA LIKE METHOD FOR THE SOLUTION OF OPTIMAL CONTROL MODEL OF REAL INVESTMENT AND FISH MANAGEMENT

This study develops the Runge-Kutta Like Method (RKLM), which uses Pontryagin's principle to solve optimal control problems numerically using forward-backward sweep methods. It is based on the Patade and Bhalekar methodology. The RKLM's stability properties and its convergence are examined. The Forward-backward sweep algorithm and the RKLM algorithm are implemented using MATLAB code. Physical optimum control problems are solved with the RKLM. The first problem's conclusion demonstrates that, when investment declines, the capital first grow to boost production before it depreciates. The outcome of the second problem demonstrates that a larger weight parameter causes the harvesting rate to reach zero more quickly and the total fish mass to reach its maximum level more quickly. The findings obtained demonstrate the effectiveness of using RKLM in conjunction with forward-backward sweep methods to solve optimal control problems.

Iterative method for the numerical solution of optimal control model for mosquito and insecticide

A linear multistep method is transformed into an iterative method based on Patade and Bhalekar's technique for the numerical solution of the optimal control problem modeled for mosquito and insecticide management using forward-backward sweep methods via Pontryagin's principle. Stability and convergence analysis of the iterative method are carried out, and it is found to be stable, convergent, and of order four. Results obtained by the method clearly show that the population of mosquitoes can be minimized to a large extent using the new iterative method while reducing the harmful effects of the insecticide, subsequently reducing the spread of malaria.

CONTROL THEORY APPROACHES TO OPTIMIZE INFORMATION FLOW IN THE FACE OF UNCERTAINTY AND CONFLICTING DATA

The rapid spread of inaccurate information through social media has become a major concern in recent years. This paper presents a mathematical model for analyzing the impact of inaccurate information on the spread of accurate information in a network. We aim to evaluate the effectiveness of three different strategies of control: media literacy programs and fact-checking and a combination of both. The model is analyzed using the maximum principle of Pontryagin to derive the optimal control strategies for minimizing the spread of inaccurate information while maximizing the spread of accurate information. A numerical simulation is presented to illustrate the effectiveness of the control strategies, and statistical analysis is performed to compare the impact of the different control strategies on the spread of information. The results demonstrate that the combination of media literacy programs and fact-checking is the most effective strategy for increasing the spread of accurate information and reducing the spread of inaccurate information. These findings have important implications for the design of effective strategies to combat the spread of misinformation and promote the spread of accurate information in a network.

Matrix Pontryagin principle approach to controllability metrics maximization under sparsity constraints

This study considers optimization problems for time-varying network topologies that maximize the controllability of linear network systems under sparsity constraints. The major theoretical differences from the existing findings on control node scheduling problems is that our method is based on the matrix-valued Pontryagin maximum principle applied to the controllability Lyapunov differential equation. This allows us to handle various controllability metrics that can only be approximated by existing methods. Finally, we perform a numerical evaluation of our proposed method and compare results with those of our previous approximation method.

Analyzing nonlinear dynamics of dengue epidemics: A fractional order model with alert state and optimal control strategies

AbstractDengue fever is a viral disease that results in numerous fatalities and considerable financial burden. Although vaccines have been developed to treat it, some countries still have endemic cases of the disease. To manage the transmission of the disease in these areas, we create a new model that includes an alert compartment to analyze the dynamics of dengue using a fractional‐order approach. We conduct a qualitative analysis of the model and use the Pontryagin principle to develop and study an optimal control problem. Since there is no definitive treatment for dengue, it is essential to prioritize measures that control or prevent the transmission of the disease. We consider control measures, including prevention efforts, insecticides, educational campaigns, and treatment for humans. We can analyze their impact on the transmission dynamics through numerical simulations. We compare our model to an existing one in two ways: with alert and without alert. We analyze the memory effect and validate the results by interpreting parameters on the reproduction number. Our results indicate that individual control measures are insufficient to eliminate the disease, and a combination of all control measures with timely alerts to the population is necessary.

STRATEGI KONTROL OPTIMAL PADA SISTEM DINAMIK PEROKOK

This study aimed to investigate optimal strategies in dynamic models of smoking. The smoking dynamic model is divided into 3 subpopulations including potential smokers (non-smokers), active smokers who smoke daily, and people who have quit smoking permanently. There are five variables of the smoking dynamic model control strategy, including education related to the dangers of smoking for health, vaccination, tobacco taxation, treatment, and rehabilitation. In solving optimization problems, this study uses the Maximum Pontryagin Principle method. Furthermore, the 4th-order runge kutta method was used to implement numerical solutions and Matlab Software to simulate a control model of smoking dynamics. Based on the simulation results, it can be seen that the control provided is effective in reducing the number of smokers and increasing the number of people who quit smoking.

Analysis and optimal control problem for a fractional mathematical model of tuberculosis with smoking consideration

This article studies a mathematical model of the fractional order of tuberculosis (TB). It describes the dynamics of the spread of tuberculosis among smokers. The purpose of this research is to protect vulnerable people against the virus. According to the survey results, the required model has an equilibrium point: the disease-free equilibrium point Ef. We also analyze the local stability of this equilibrium point of the model, using the basic reproduction number R0 calculated according to the new generation method. In our model, we include three controls that represent: restricting individual contact, treatment, and sensitization. This article aims at reducing the number of infected smokers and non-smokers using an optimal control strategy and a fractional derivation. The maximum principle of Pontryagin is used to describe optimal controls with Caputo-derived fractional over time and the optimal system is resolved iteratively. The numerical simulation is presented according to the method presented by Matlab.

Optimal control dynamics of Gonorrhea in a structured population

Gonorrhea is a serious global health problem due to its high incidence, with approximately 82.4 million new cases in 2020. To evaluate the consequences of targeted dynamic control of gonorrhea infection transmission, a model for gonorrhea with optimal control analysis is proposed for a structured population. The study looked at the model's positively invariant and bounded regions. The gonorrhea secondary infection expression, R0 for the structured population is computed. The maximum principle of Pontryagin is utilised to construct the optimal system for the formulated mathematical model. To reduce the continuous propagation of gonorrhea, we incorporated education, condoms usage, vaccinations, and treatment as control strategies. The numerical simulations show that the number of infections decreases when the controls are implemented. The effectiveness of the controls is shown using the efficacy plots.

Pontryagin's principle and variational inequality for optimal control problems governed by viscous Camassa–Holm equations

In this paper we prove the existence of optimal solutions as well as first-order necessary optimality conditions for a distributed optimal control problem governed by the three-dimensional viscous Camassa–Holm equations in bounded domains with the cost function of a quite general form. Particularly, in the box constraints case, the connection between the first-order necessary optimality conditions in the variational inequality form and the Pontryagin minimum principle is also established.

Dynamic Analysis and Optimal Control of Fractional Order African Swine Fever Models with Media Coverage.

African swine fever is a highly contagious virus that causes pig disease. Its onset process is short, but the mortality rate is as high as 100%. There are still no effective drugs that have been developed to treat African swine fever, and prevention and control measures are currently the best means to avoid infection in pig herds. In this paper, two fractional order mathematical models with media coverage are constructed to describe the transmission of African swine fever. The first model is a basic model with media coverage, and no control measures are considered. For this model, the reproduction number is obtained by using the next generation matrix method. Then, the sufficient conditions for the existence and stability of two equilibriums are obtained. Based on the first model, the second model is established incorporating two control measures. By using Pontryagin's maximal principle, the optimal control solution is derived. After that, some numerical simulations are performed for the two models to verify the theoretical results. Both the qualitative analysis and numerical results indicate that timely media coverage combined with disinfection control measures is crucial to preventing the spread of disease.

Structure-preserving integrators based on a new variational principle for constrained mechanical systems

A new variational principle for mechanical systems subject to holonomic constraints is presented. The newly proposed GGL principle is closely related to the often used Gear-Gupta-Leimkuhler (GGL) stabilization of the differential–algebraic equations governing the motion of constrained mechanical systems. The GGL variational principle relies on an extension of the Livens principle (sometimes also referred to as Hamilton–Pontryagin principle) to mechanical systems subject to holonomic constraints. In contrast to the original GGL stabilization, the new approach facilitates the design of structure-preserving integrators. In particular, new variational integrators are presented, which result from the direct discretization of the GGL variational principle. These variational integrators are symplectic and conserve momentum maps in the case of systems with symmetry. In addition to that, a new energy–momentum scheme is developed, which results from the discretization of the Euler–Lagrange equations pertaining to the GGL variational principle. The numerical properties of the newly devised schemes are investigated in representative examples of constrained mechanical systems.

Importance sampling for McKean-Vlasov SDEs

This paper deals with Monte-Carlo (MC) methods for evaluating expectations of functionals of solutions to McKean-Vlasov Stochastic Differential Equations (MV-SDE) including those with super-linearly growing drifts. Underpinned by an interacting particle system approximation, we propose two importance sampling (IS) algorithms to reduce the variance of an associated MC estimator.The “complete measure change” algorithm sees the IS measure change applied to the target expectation to be calculated and to the MV-SDE coefficients simultaneously. The “decoupling algorithm” consists of estimating the law of the MV-SDE’s solution via standard simulation under the initial measure, then fixing the law component of the MV-SDE via that simulation, and finally simulating the new equation under the IS measure.Methodologically, large deviations and Pontryagin principle are employed to determine and solve the variance minimisation problem that yields the required measure change. The optimisation problem associated to the complete measure change is more complex than that for the decoupling algorithm, nonetheless, symmetry arguments allow for non-trivial complexity reduction. As an example, both algorithms are tested using the Kuramoto model from statistical physics. For the functionals tested, we see a reduction of up to 3 orders of magnitude on the variance of both IS schemes in comparison to the standard Monte Carlo approximation. In terms of computational cost, the complete measure change is akin to standard Monte Carlo whilst the decoupled approach increases the cost by a factor of around 2 if one uses the same number of particles for both steps. The statistical error of the method dominates the propagation of chaos error by 1 order of magnitude.

On p-adic tight wavelet frames

We propose a simple method to constructing step mask and corresponding step wavelet functions that generate tight wavelet frames on the field Qp of p-adic numbers. To construct this mask we use three new ideas. First, we consider only an additive group of the field Qp. Second, to construct the mask we use a special tree. Thirdly, we do not use the principle of unitary extension, we use Pontryagin's principle of duality instead. Using this method, we obtain all step refinable functions.

Optimal control of a Wilson-Cowan model of neural population dynamics.

Nonlinear dynamical systems describe neural activity at various scales and are frequently used to study brain functions and the impact of external perturbations. Here, we explore methods from optimal control theory (OCT) to study efficient, stimulating "control" signals designed to make the neural activity match desired targets. Efficiency is quantified by a cost functional, which trades control strength against closeness to the target activity. Pontryagin's principle then enables to compute the cost-minimizing control signal. We then apply OCT to a Wilson-Cowan model of coupled excitatory and inhibitory neural populations. The model exhibits an oscillatory regime, low- and high-activity fixed points, and a bistable regime where low- and high-activity states coexist. We compute an optimal control for a state-switching (bistable regime) and a phase-shifting task (oscillatory regime) and allow for a finite transition period before penalizing the deviation from the target state. For the state-switching task, pulses of limited input strength push the activity minimally into the target basin of attraction. Pulse shapes do not change qualitatively when varying the duration of the transition period. For the phase-shifting task, periodic control signals cover the whole transition period. Amplitudes decrease when transition periods are extended, and their shapes are related to the phase sensitivity profile of the model to pulsed perturbations. Penalizing control strength via the integrated 1-norm yields control inputs targeting only one population for both tasks. Whether control inputs drive the excitatory or inhibitory population depends on the state-space location.

Analisis dan Kontrol Optimal pada Model Dinamik Penyebaran Virus Zika

Zika is a virus from flaviviridae family and flavivirus genus that spread by Aedes Aegypti mosquito and that can cause serious problem disease such as Guillain Barre Syndrome (GBS) and mikrosefalus. In this paper used spreading dynamic zika virus model that consists of two population and divided to sub population Susceptible humans, sub-population Asymtomatic Infected humans, sub-population Infected humans, sub-population Recovered humans, sub-population Susceptible mosquitoes, and sub-population Infected mosquitoes. The model that anlysis by determine the basic reproduction number, the point of disease free and endemic equilibrium, and the stability of each point of equilibrium. Then, do the optimal control using Pontryagin principle with numerical solution given by Range-Kutta method. The simulation results show the decreasing sub-population of infected humans, asymptomatic infected human, and population of mosquitoes after given controls such using condom, treatment, and using indoor residual spray.