Pontryagin’s Maximum Principle for a Advection-Diffusion-Reaction Equation

In this paper we investigate optimal control problems governed by a advection-diffusion-reaction equation. We present a method for deriving conditions in the form of Pontryagin’s principle. The main tools used are the Ekeland’s variational principle combined with penalization and spike variation techniques.

基于函数S-粗集的Bellman原理优化算法在知识测度的应用

应用函数双向S-粗集理论实现参考模式和测试模式的动态模式匹配,用Bellman原理的动态规划算法实现全局约束定义下的Levenstein距离的计算,以此确定出参考模式和测试模式的距离测度,有明显降低计算复杂度的效果,并以无纸考试系统非标准化试题的智能评分为例进行说明。

Parametric Iteration Method for Solving Linear Optimal Control Problems

This article presents the Parametric Iteration Method (PIM) for finding optimal control and its corresponding trajectory of linear systems. Without any discretization or transformation, PIM provides a sequence of functions which converges to the exact solution of problem. Our emphasis will be on an auxiliary parameter which directly affects on the rate of convergence. Comparison of PIM and the Variational Iteration Method (VIM) is given to show the preference of PIM over VIM. Numerical results are given for several test examples to demonstrate the applicability and efficiency of the method.

Modeling and Numerical Solution of a Cancer Therapy Optimal Control Problem

A mathematical optimal-control tumor therapy framework consisting of radio- and anti-angiogenesis control strategies that are included in a tumor growth model is investigated. The governing system, resulting from the combination of two well established models, represents the differential constraint of a non-smooth optimal control problem that aims at reducing the volume of the tumor while keeping the radio- and anti-angiogenesis chemical dosage to a minimum. Existence of optimal solutions is proved and necessary conditions are formulated in terms of the Pontryagin maximum principle. Based on this principle, a so-called sequential quadratic Hamiltonian (SQH) method is discussed and benchmarked with an “interior point optimizer—a mathematical programming language” (IPOPT-AMPL) algorithm. Results of numerical experiments are presented that successfully validate the SQH solution scheme. Further, it is shown how to choose the optimisation weights in order to obtain treatment functions that successfully reduce the tumor volume to zero.

Modelling the Optimal Control of Transmission Dynamics of <i>Mycobacterium ulceran</i>Infection

This paper examines optimal control of transmission dynamics of Mycobacterium ulceran (MU) infection. A nonlinear mathematical model for the problem is proposed and analysed qualitatively using the stability theory of the differential equations, optimal control and computer simulation. The basic reproduction number of the reduced model system is obtained by using the next generation operator method. It is found that by using Ruth Hurwitz criteria, the disease free equilibrium point is locally asymptotically stable and using centre manifold theory, the model shows the transcritical (forward) bifurcation. Optimal control is applied to the model seeking to minimize the transmission dynamics of MU infection on human and water-bugs. Pontryagin’s maximum principle is used to characterize the optimal levels of the controls. The results of optimality are solved numerically using MATLAB software and the results show that optimal combination of two controls (environmental and health education for prevention) and (water and environmental purification) minimizes the MU infection in the population.

Analysis of Listeriosis Transmission Dynamics with Optimal Control

Listeriosis is an illness caused by the germ Listeria monocytogenes. Generally, humans are infected with listeriosis after eating contaminated food. Listeriosis mostly affects people with weakened immune systems, pregnant women and newborns. In this paper, a model describing the dynamics of Listeriosis is developed and analysed using ordinary differential equations. The model was analysed both quantitatively and qualitatively for its local and global stability, basic reproductive number and parameter contributions to the basic reproductive number to understand the impact of each parameter on the disease spread. The Listeriosis model has been extended to include time dependent control variables such as treatment of both humans and animals, vaccination and education of humans. Pontryagin’s Maximum Principle was introduced to obtain the best optimal control strategies required for curbing Listeriosis infections. Numerical simulation was performed and the results displayed graphically and discussed. Cost effectiveness analysis was conducted using the intervention averted ratio (IAR) concepts and it was revealed that the most effective intervention strategy is the treatment of infected humans and animals.

Mathematical Analysis of Nipah Virus Infections Using Optimal Control Theory

The optimal use of intervention strategies to mitigate the spread of Nipah Virus (NiV) using optimal control technique is studied in this paper. First of all we formulate a dynamic model of NiV infections with variable size population and two control strategies where creating awareness and treatment are considered as controls. We intend to find the optimal combination of these two control strategies that will minimize the cost of the two control measures and as a result the number of infectious individuals will decrease. We establish the existence for the optimal controls and Pontryagin’s maximum principle is used to characterize the optimal controls. The numerical simulation suggests that optimal control technique is much more effective to minimize the infected individuals and the corresponding cost of the two controls. It is also monitored that in the case of high contact rate, controls have to work for longer period of time to get the desired result. Numerical simulation reveals that the spread of Nipah virus can be controlled effectively if we apply control strategy at early stage.

Some Applications of Optimal Control in Sustainable Fishing in the Baltic Sea

Issues related to the implementation of dynamic programming for optimal control of a three-dimensional dynamic model (the fish populations management problem) are presented. They belong to a class of models called Lotka-Volterra models. The existence of bionomic equilibria will be considered. The problem of optimal harvest policy is then solved for the control of various classes of its behaviour. Therefore the focus will be the optimality conditions by using the Bellman principle. Moreover, we consider a different form for the optimal value of the control vector, namely the feedback or closed-loop form of the control. Academic examples are studied in order to demonstrate the proposed methods.

Study of optimal control problems for hybrid dynamical systems

From the viewpoint of continuous systems, optimal control problem is proposed for a class of controlled Hybrid dynamical systems. Then a mathematical method- HDS minimum principle is put forward, which can solve the above problem. The HDS minimum principle is proved by means of Ekeland＇ s variational principle.

Optimal Campaign in Leptospirosis Epidemic by Multiple Control Variables

In this paper, we consider a leptospirosis epidemic model to implement optimal campaign by using multiple control variables. First, we show the existence of the control problem. Then we derive the conditions under which it is optimal to eradicate the leptospirosis infection and examine the impact of a possible educatioal/vaccinaction campaign using Pontryagin’s Maximum Principle. We completely characterize the optimal control problem and compute the numerical solution of the optimality system using an iterative method. The results obtained from the numerical simulations of the model show that a possible educational/vaccinaction combined with effective treatment regime would reduce the spread of the leptospirosis infection appreciably.

Optimal Control of Heterogeneous-Susceptible-Exposed-Infectious-Recovered-Susceptible Malware Propagation Model in Heterogeneous Degree-Based Wireless Sensor Networks

Heterogeneous wireless sensor networks(HWSNs)are vulnerable to malware propagation,because of their low configuration and weak defense mechanism.Therefore,an optimality system for HWSNs is developed to suppress malware propagation in this paper.Firstly,a heterogeneous-susceptible-exposed-infectious-recovered-susceptible(HSEIRS)model is proposed to describe the state dynamics of heterogeneous sensor nodes(HSNs)in HWSNs.Secondly,the existence of an optimal control problem with installing antivirus on HSNs to minimize the sum of the cumulative infection probabilities of HWSNs at a low cost based on the HSEIRS model is proved,and then an optimal control strategy for the problem is derived by the optimal control theory.Thirdly,the optimal control strategy based on the HSEIRS model is transformed into corresponding Hamiltonian by the Pontryagin’s minimum principle,and the corresponding optimality system is derived.Finally,the effectiveness of the optimality system is validated by the experimental simulations,and the results show that the infectious HSNs will fall to an extremely low level at a low cost.

Optimal Control Analysis of Influenza Epidemic Model

The implementation of optimal control strategies involving preventive measures and antiviral treatment can significantly reduce the number of clinical cases of influenza. In this paper, a model for the transmission dynamics of influenza is formulated and two control strategies involving preventive measures (awareness campaign, washing hand, using hand sanitizer, wearing mask) and treatment are considered and used to minimize the total number of infected individuals and associated cost of using these two controls. The resulting optimality system is solved numerically. Hamiltonian is formulated to investigate the existence of the optimal control, in the optimal control model. Pontryagin’s Maximum Principle is applied to describe the control variables and the objective function is designed to reduce both the infection and the cost of interventions. From the numerical simulation, it is observed that in the case of high contact rate (β = 3), both the controls work for a longer period of time to reduce the disease burden. The optimal control analysis and numerical simulations reveal that the interventions reduce the number of exposed and infected individuals.

Neural Network Approach for Solving Singular Convex Optimization with Bounded Variables

Although frequently encountered in many practical applications, singular nonlinear optimization has been always recognized as a difficult problem. In the last decades, classical numerical techniques have been proposed to deal with the singular problem. However, the issue of numerical instability and high computational complexity has not found a satisfactory solution so far. In this paper, we consider the singular optimization problem with bounded variables constraint rather than the common unconstraint model. A novel neural network model was proposed for solving the problem of singular convex optimization with bounded variables. Under the assumption of rank one defect, the original difficult problem is transformed into nonsingular constrained optimization problem by enforcing a tensor term. By using the augmented Lagrangian method and the projection technique, it is proven that the proposed continuous model is convergent to the solution of the singular optimization problem. Numerical simulation further confirmed the effectiveness of the proposed neural network approach.

基于Bellman原理的人员密集场所疏散路径优化研究

具有多决策点的人员密集场所存在多条疏散路径,合理规划疏散路径可以避免长时间的局部拥堵,降低踩踏事故发生概率。人员应急疏散仿真工程软件Pathfinder的局部最快算法不能解决多步决策场所中路径利用率不均衡的问题,为此提出一种以总疏散时间最短为目标的路径优化方法,基于决策点构建可行路径模型及路径优化模型,基于Bellman原理对优化模型进行求解,得到较优路径方案。依照该方案推演疏散过程,更新较优路径直至重复多次无更优路径出现,最终确定最优路径及其疏散时间。以某学生公寓人员疏散为例论证该方案的优化效果,结果表明与Pathfinder模拟结果相比,该方法总疏散时间从330.3 s减少至292.3 s,东、西侧一楼楼梯出口的利用时间差从75 s缩减为1.5 s,通过人数差从138人降为0人。该路径优化方法可用于改进疏散模拟软件。

一类具有Beddington-DeAngelis功能反应捕食系统收获模型的稳定性

研究一类具有双线性密度制约及Beddington-DeAngelis功能反应的捕食系统收获模型.运用微分方程定性稳定性理论讨论系统正平衡点的性态,得到其局部渐近稳定及全局渐近稳定的充分条件,利用Pontryagin最大值原理得到系统的最优收获策略.

带Markov跳的离散时间随机控制系统的最大值原理

本文研究一类同时含有Markov跳过程和乘性噪声的离散时间非线性随机系统的最优控制问题,给出并证明了相应的最大值原理.首先,利用条件期望的平滑性,通过引入具有适应解的倒向随机差分方程,给出了带有线性差分方程约束的线性泛函的表示形式,并利用Riesz定理证明其唯一性.其次,对带Markov跳的非线性随机控制系统,利用针状变分法,对状态方程进行一阶变分,获得其变分所满足的线性差分方程.然后,在引入Hamilton函数的基础上,通过一对由倒向随机差分方程刻画的伴随方程,给出并证明了带有Markov跳的离散时间非线性随机最优控制问题的最大值原理,并给出该最优控制问题的一个充分条件和相应的Hamilton-Jacobi-Bellman方程.最后,通过一个实际例子说明了所提理论的实用性和可行性.

Motoyosi Sugita—A “Widely Unknown” Japanese Thermodynamicist Who Explored the 4th Law of Thermodynamics for Creation of the Theory of Life

The purpose of this paper is to introduce to you, the Western people, nowadays a “widely unknown” Japanese thermodynamicist by the name of Motoyosi Sugita and his study on the thermodynamics of transient phenomena and his theory of life. This is because although he was one of the top theoretical physicists in Japan before, during and after WWII and after WWII he promoted the establishment of the biophysical society of Japan as one of the founding members, he himself and his studies themselves have seemed to be totally forgotten nowadays in spite that his study was absolutely important for the study of life. Therefore, in this paper I would like to present what kind of person he was and what he studied in physics as a review on the physics work of Motoyosi Sugita for the first time. I will follow his past studies to introduce his ideas in theoretical physics as well as in biophysics as follows: He proposed the bright ideas such as the quasi-static change in the broad sense, the virtual heat, and the field of chemical potential etc. in order to establish his own theory of thermodynamics of transient phenomena, as the generalization of the Onsager-Prigogine’s theory of the irreversible processes. By the concept of the field of chemical potential that acquired the nonlinear transport, he was seemingly successful to exceed and go beyond the scope of Onsager and Prigogine. Once he established his thermodynamics, he explored the existence of the 4th law of thermodynamics for the foundation of theory of life. He applied it to broad categories of transient phenomena including life and life being such as the theory of metabolism. He regarded the 4th law of thermodynamics as the maximum principle in transient phenomena. He tried to prove it all life long. Since I have recently found that his maximum principle can be included in more general maximum principle, which was known as the Pontryagin’s maximum principle in the theory of optimal control, I would like to explain such theories produced by Motoyosi Sugita as detailed as possible. And also I have put short history of Motoyosi Sugita’s personal life in order for you to know him well. I hope that this article helps you to know this wonderful man and understand what he did in the past, which was totally forgotten in the world and even in Japan.

具有年龄结构的SIR传染病模型的最优接种和治疗策略

研究了一类染病者具有年龄结构的SIR传染病模型的最优接种和治疗策略问题.利用Banach压缩映射原理和Gronwall引理,证明了该传染病模型非负解的唯一性以及解对控制变量的连续依赖性.借助切锥法锥技巧给出最优接种和治疗策略的必要条件.根据Ekeland变分原理确立了最优接种和治疗策略的存在性和唯一性.

中立型随机时滞微分方程最优性的Bellman原则(英文)

考虑一类随机中立型时滞微分方程最优性的Bellm an原则问题.推广了随机微分方程和随机时滞微分方程的相应结论.

具有Michaelis-Menten收获率的捕食者-食饵征税模型的研究

研究一类具有Michaelis-Menten收获率的Holling-III类功能性捕食者-食饵征税模型,利用微分方程定性理论,讨论正平衡点的局部渐近稳定性,通过构造Lyapunov函数,给出正平衡点全局渐近稳定性的充分条件。利用Pontryagain最大值原理,确定最优征税策略。用Matlab软件进行模拟,验证了正平衡点稳定性和最优税收策略的理论结果。