scholarly journals Mathematical Analysis and Optimal Control of Giving up the Smoking Model

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Omar Khyar ◽  
Jaouad Danane ◽  
Karam Allali
Keyword(s):  
Optimal Control ◽  
Mathematical Analysis ◽  
Minimum Principle ◽  
Smoking Behavior ◽  
Difference Approximation ◽  
Optimal Controls ◽  
Pontryagin Minimum Principle ◽  
Control Pair ◽  
Public Areas ◽  
The Government

In this study, we are going to explore mathematically the dynamics of giving up smoking behavior. For this purpose, we will perform a mathematical analysis of a smoking model and suggest some conditions to control this serious burden on public health. The model under consideration describes the interaction between the potential smokers P , the occasional smokers L , the chain smokers S , the temporarily quit smokers Q T , and the permanently quit smokers Q P . Existence, positivity, and boundedness of the proposed problem solutions are proved. Local stability of the equilibria is established by using Routh–Hurwitz conditions. Moreover, the global stability of the same equilibria is fulfilled through using suitable Lyapunov functionals. In order to study the optimal control of our problem, we will take into account a two controls’ strategy. The first control will represent the government prohibition of smoking in public areas which reduces the contact between nonsmokers and smokers, while the second will symbolize the educational campaigns and the increase of cigarette cost which prevents occasional smokers from becoming chain smokers. The existence of the optimal control pair is discussed, and by using Pontryagin minimum principle, these two optimal controls are characterized. The optimality system is derived and solved numerically using the forward and backward difference approximation. Finally, numerical simulations are performed in order to check the equilibria stability, confirm the theoretical findings, and show the role of optimal strategy in controlling the smoking severity.

scholarly journals Optimal Control of a Delayed HIV Infection Model with Immune Response Using an Efficient Numerical Method

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
Khalid Hattaf ◽  
Noura Yousfi
Keyword(s):  
Optimal Control ◽  
Immune Response ◽  
Hiv Infection ◽  
Equation Model ◽  
Difference Approximation ◽  
Infection Model ◽  
Optimal Controls ◽  
Control Pair ◽  
Hiv Infection Model ◽  
Delay Differential

We present a delay-differential equation model with optimal control that describes the interactions between human immunodeficiency virus (HIV), CD4+ T cells, and cell-mediated immune response. Both the treatment and the intracellular delay are incorporated into the model in order to improve therapies to cure HIV infection. The optimal controls represent the efficiency of drug treatment in inhibiting viral production and preventing new infections. Existence for the optimal control pair is established, Pontryagin’s maximum principle is used to characterize these optimal controls, and the optimality system is derived. For the numerical simulation, we propose a new algorithm based on the forward and backward difference approximation.

scholarly journals Pontryagin Neural Networks with Functional Interpolation for Optimal Intercept Problems

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 996
Author(s):  
Andrea D’Ambrosio ◽  
Enrico Schiassi ◽  
Fabio Curti ◽  
Roberto Furfaro
Keyword(s):  
Neural Networks ◽  
Optimal Control ◽  
Minimum Principle ◽  
Necessary Optimality Conditions ◽  
Optimal Controls ◽  
Pontryagin Minimum Principle ◽  
First Order ◽  
Functional Connections ◽  
Control Actions ◽  
Interpolation Technique

In this work, we introduce Pontryagin Neural Networks (PoNNs) and employ them to learn the optimal control actions for unconstrained and constrained optimal intercept problems. PoNNs represent a particular family of Physics-Informed Neural Networks (PINNs) specifically designed for tackling optimal control problems via the Pontryagin Minimum Principle (PMP) application (e.g., indirect method). The PMP provides first-order necessary optimality conditions, which result in a Two-Point Boundary Value Problem (TPBVP). More precisely, PoNNs learn the optimal control actions from the unknown solutions of the arising TPBVP, modeling them with Neural Networks (NNs). The characteristic feature of PoNNs is the use of PINNs combined with a functional interpolation technique, named the Theory of Functional Connections (TFC), which forms the so-called PINN-TFC based frameworks. According to these frameworks, the unknown solutions are modeled via the TFC’s constrained expressions using NNs as free functions. The results show that PoNNs can be successfully applied to learn optimal controls for the class of optimal intercept problems considered in this paper.

scholarly journals Application and benchmarking of a direct method to optimize the fuel consumption of a diesel electric locomotive

2018 ◽  
Vol 232 (9) ◽  
pp. 2272-2289 ◽  
Author(s):  
V Macian ◽  
C Guardiola ◽  
B Pla ◽  
A Reig
Keyword(s):  
Optimal Control ◽  
Direct Method ◽  
Minimum Principle ◽  
Optimization Technique ◽  
Direct Methods ◽  
Optimization Methods ◽  
Computational Time ◽  
Electric Locomotive ◽  
Pontryagin Minimum Principle ◽  
A Direct Method

This paper addresses the optimal control of a long-haul passenger train to deliver minimum-fuel operations. Contrary to the common Pontryagin minimum principle approach in railroad-related literature, this work addresses this optimal control problem with a direct method of optimization, the use of which is still marginal in this field. The implementation of a particular direct method based on the Euler collocation scheme and its transcription into a nonlinear problem are described in detail. In this paper, this optimization technique is benchmarked with well-known optimization methods in the literature, namely dynamic programming and the Pontryagin minimum principle, by simulating a real route. The results showed that the direct methods are on the same level of optimality compared with other algorithms while requiring reduced computational time and memory and being able to handle very complex dynamic systems. The performance of the direct method is also compared to the real trajectory followed by the train operator and exhibits up to 20% of fuel saving in the example route.

scholarly journals State-Based Switching for Optimal Control of Computer Virus Propagation with External Device Blocking

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Qingyi Zhu ◽  
Seng W. Loke ◽  
Ye Zhang
Keyword(s):  
Optimal Control ◽  
Control Strategies ◽  
Minimum Principle ◽  
Optimal Solution ◽  
Computer Virus ◽  
Numerical Results ◽  
Difference Approximation ◽  
External Device ◽  
Virus Propagation ◽  
The Impact

The rapid propagation of computer virus is one of the greatest threats to current cybersecurity. This work deals with the optimal control problem of virus propagation among computers and external devices. To formulate this problem, two control strategies are introduced: (a) external device blocking, which means prohibiting a fraction of connections between external devices and computers, and (b) computer reconstruction, which includes updating or reinstalling of some infected computers. Then the combination of both the impact of infection and the cost of controls is minimized. In contrast with previous works, this paper takes into account a state-based cost weight index in the objection function instead of a fixed one. By using Pontryagin’s minimum principle and a modified forward-backward difference approximation algorithm, the optimal solution of the system is investigated and numerically solved. Then numerical results show the flexibility of proposed approach compared to the regular optimal control. More numerical results are also given to evaluate the performance of our approach with respect to various weight indexes.

scholarly journals Prospects on Solving an Optimal Control Problem with Bounded Uncertainties on Parameters using Interval Arithmetics

2021 ◽  
Author(s):  
Etienne Bertin ◽  
Elliot Brendel ◽  
Bruno Hérissé ◽  
Julien Alexandre dit Sandretto ◽  
Alexandre Chapoutot
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Closed Loop ◽  
Minimum Principle ◽  
Open Loop ◽  
Pontryagin Minimum Principle ◽  
Interval Method ◽  
Bounded Uncertainties ◽  
The Given

An interval method based on the Pontryagin Minimum Principle is proposed to enclose the solutions of an optimal control problem with embedded bounded uncertainties. This method is used to compute an enclosure of all optimal trajectories of the problem, as well as open loop and closed loop enclosures meant to enclose a concrete system using an optimal control regulator with inaccurate knowledge of the parameters. The differences in geometry of these enclosures are exposed, as well as some applications. For instance guaranteeing that the given optimal control problem will yield a satisfactory trajectory for any realization of the uncertainties or on the contrary that the problem is unsuitable and needs to be adjusted.

An Optimal Control Problem of Knowledge Dissemination

2020 ◽  
pp. 461-482
Author(s):  
Pantry Elastic ◽  
Toni Bakhtiar ◽  
Jaharuddin
Keyword(s):  
Optimal Control ◽  
Information Dissemination ◽  
Control Strategies ◽  
Minimum Principle ◽  
Control Model ◽  
Technical Training ◽  
Knowledge Dissemination ◽  
Transfer System ◽  
Pontryagin Minimum Principle

In this chapter, the authors develop an optimal control model of knowledge dissemination among people in the society. The knowledge transfer system is formulated in term of compartmental model, where the society members are categorized into four classes based on knowledge acquisition and their willingness to disseminate. The model is equipped with a set of control variables for process intervening, namely technical training for ignorant-immigrants, information dissemination through social media for solitariants and enthusiants, and technical training for solitariants. Optimality conditions in terms of differential equations system was derived by using Pontryagin minimum principle leading to the characterization of optimal control strategies that minimizing the number of solitariants, enthusiants, and ignorants simultaneously with the control efforts. The sweep method and the fourth order Runge-Kutta algorithm was implemented to numerically solve the equation systems. The effectiveness of the control strategies toward a set of control scenarios was evaluated through examples.

scholarly journals Mathematical Analysis and Treatment for a Delayed Hepatitis B Viral Infection Model with the Adaptive Immune Response and DNA-Containing Capsids

2018 ◽  
Vol 7 (4) ◽  
pp. 35 ◽  
Author(s):  
Jaouad Danane ◽  
Karam Allali
Keyword(s):  
Hepatitis B ◽  
Cytotoxic T Lymphocyte ◽  
Minimum Principle ◽  
Adaptive Immune Response ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Infection Model ◽  
Control Pair ◽  
B Virus ◽  
Hepatitis B Viral Infection

We model the transmission of the hepatitis B virus (HBV) by six differential equations that represent the reactions between HBV with DNA-containing capsids, the hepatocytes, the antibodies and the cytotoxic T-lymphocyte (CTL) cells. The intracellular delay and treatment are integrated into the model. The existence of the optimal control pair is supported and the characterization of this pair is given by the Pontryagin’s minimum principle. Note that one of them describes the effectiveness of medical treatment in restraining viral production, while the second stands for the success of drug treatment in blocking new infections. Using the finite difference approximation, the optimality system is derived and solved numerically. Finally, the numerical simulations are illustrated in order to determine the role of optimal treatment in preventing viral replication.

scholarly journals Optimal Control of Mathematical Models on The Dynamics Spread of Drug Abuse

2021 ◽  
Vol 17 (3) ◽  
pp. 339-348
Author(s):  
Nita Anggriani ◽  
Syamsuddin Toaha ◽  
Kasbawati Kasbawati
Keyword(s):  
Mathematical Model ◽  
Optimal Control ◽  
Drug Abuse ◽  
Mathematical Models ◽  
Minimum Principle ◽  
Self Psychology ◽  
Sweep Method ◽  
Pontryagin Minimum Principle ◽  
Counseling Needs ◽  
Forward Backward Sweep Method

This article examines the optimal control of a mathematical model of the spread of drug abuse. This model consists of five population classes, namely susceptible to using drugs (S), light-grade drugs (A), heavy-grade drugs (H), medicated drugs (T), and Recovery from drugs (R). The system is solved using the Pontryagin minimum principle and numerically by the forward-backward sweep method. Numerical simulations of the optimal problem show that with the implementation of anti-drug campaigns and strengthening of self-psychology through counseling, the spread of drug abuse can be eradicated more quickly. The implementation of campaigns and strengthening of self-psychology through large amounts of counseling needs to be done from the beginning then the proportion can be reduced until a certain time does not need to be given anymore. The use of control in the form of strengthening efforts to self-psychology through counseling means that it needs to be done in a longer time to prevent the spread of drug abuse.

scholarly journals KONTROL OPTIMAL MODEL PENYEBARAN VIRUS KOMPUTER DENGAN PENGARUH KOMPUTER EKSTERNAL YANG TERINFEKSI DAN REMOVABLE STORAGE MEDIA

2017 ◽  
Vol 9 (1) ◽  
pp. 113
Author(s):  
Dewi Erla Mahmudah ◽  
Muhammad Zidny Naf’an ◽  
Muh. Sofi’i ◽  
Wika Wika
Keyword(s):  
Optimal Control ◽  
Optimality Condition ◽  
Minimum Principle ◽  
Prevention Strategies ◽  
Optimal Model ◽  
Pontryagin Minimum Principle ◽  
Computer Viruses ◽  
Storage Media ◽  
The Cost

.  In this paper, we discuss an optimal control on the spread of computer viruses under the effects of infected external computers and removable storage media. Prevention Strategies do with ascertaining control prevention to minimize the number of infective computers (Latent and Breakingout) and installing effective antivirus programs in each sub-population. The aim are to derive optimal prevention strategies and minimize the cost associated with the control. The characterization of optimal control is perform analitically by applying Pontryagin Minimum Principle. The obtained optimality system of Hamilton fuction is satistfy the optimality condition.

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