一类具年龄结构的线性周期种群动力系统的最优控制

研究一类具有年龄结构的线性周期种群线性动力系统的最优控制问题,即讨论了具有周期的生死率和周期变化的控制项的模型．利用Mazur's定理,证明了最优控制问题最优解的存在性,同时由法锥概念的特征刻画,我们还得到了最优控制问题最优解存在的必要条件．     

On boundary stimulation and optimal boundary control of the bidomain equations

The bidomain equations with Neumann boundary stimulation and optimal control of these stimuli are investigated. First an analytical framework for boundary control is provided. Then a parallel finite element based algorithm is devised and its efficiency is demonstrated not only for the direct problem but also for the optimal control problem. The computations realize a model configuration corresponding to optimal boundary defibrillation of a reentry phenomenon by applying current density stimuli.     

年龄相关的种群系统的最优生育率控制

讨论了一类与年龄相关的时变种群系统最优生育率控制的非线性问题,利用Ba- nach空间的Saks-Masur引理,证明了系统最优生育率控制的存在唯一性,并得到了控制为最优的必要条件和最优性组．所得结果可为种群系统控制问题的实际研究提供必要的数学理论基础．     

Allee Effect and Control of Lake System Invasion

We consider the model of invasion prevention in a system of lakes that are connected via traffic of recreational boats. It is shown that in presence of an Allee effect, the general optimal control problem can be reduced to a significantly simpler stationary optimization problem of optimal invasion stopping. We consider possible values of model parameters for zebra mussels. The general N-lake control problem has to be solved numerically, and we show a number of typical features of solutions: distribution of control efforts in space and optimal stopping configurations related with the clusters in lake connection structure.     

Optimal HIV treatment by maximising immune response

We present an optimal control model of drug treatment of the human immunodeficiency virus (HIV). Our model is based upon ordinary differential equations that describe the interaction between HIV and the specific immune response as measured by levels of natural killer cells. We establish stability results for the model. We approach the treatment problem by posing it as an optimal control problem in which we maximise the benefit based on levels of healthy CD4+ T cells and immune response cells, less the systemic cost of chemotherapy. We completely characterise the optimal control and compute a numerical solution of the optimality system via analytic continuation.Research supported by the Natural Science and Engineering Research Council (NSERC) and the Mathematics of Information Technology and Complex Systems (MITACS) of Canada     

Optimal Dynamic Allocation of Conservation Funding Among Priority Regions

The optimal allocation of conservation resources between biodiverse conservation regions has generally been calculated using stochastic dynamic programming, or using myopic heuristics. These solutions are hard to interpret and may not be optimal. To overcome these two limitations, this paper approaches the optimal conservation resource allocation problem using optimal control theory. A solution using Pontryagin’s maximum principle provides novel insight into the general properties of efficient conservation resource allocation strategies, and allows more extensive testing of the performance of myopic heuristics. We confirmed that a proposed heuristic (minimize short-term loss) yields near-optimal results in complex allocation situations, and found that a qualitative allocation feature observed in previous analyses (bang-bang allocation) is a general property of the optimal allocation strategy.     

Optimization of harvesting age in an integral age-dependent model of population dynamics

The paper deals with optimal control in a linear integral age-dependent model of population dynamics. A problem for maximizing the harvesting return on a finite time horizon is formulated and analyzed. The optimal controls are the harvesting age and the rate of population removal by harvesting. The gradient and necessary condition for an extremum are derived. A qualitative analysis of the problem is provided. The model shows the presence of a zero-investment period. A preliminary asymptotic analysis indicates possible turnpike properties of the optimal harvesting age. Biological interpretation of all results is provided.     

Optimal risk management of human alveolar echinococcosis with vermifuge

In this study, we develop a bioeconomic model of human alveolar echinococcosis (HAE) and formulate the optimal strategies for managing the infection risks in humans by applying optimal control theory. The model has the following novel features: (i) the complex transmission cycle of HAE has been tractably incorporated into the framework of optimal control problems and (ii) the volume of vermifuge spreading to manage the risk is considered a control variable. With this model, we first obtain the stability conditions for the transmission dynamics under the condition of constant control. Second, we explicitly introduce a control variable of vermifuge spreading into the analysis by considering the associated control costs. In this optimal control problem, we have successfully derived a set of conditions for a bang-bang control and singular control, which are mainly characterized by the prevalence of infection in voles and foxes and the remaining time of control. The analytical results are demonstrated by numerical analysis and we discuss the effects of the parameter values on the optimal strategy and the transmission cycle. We find that when the prevalence of infection in foxes is low and the prevalence of infection in voles is sufficiently high, the optimal strategy is to expend no effort in vermifuge spreading.     

On optimal delivery of combination therapy for tumors

A mathematical model for the scheduling of angiogenic inhibitors in combination with a chemotherapeutic agent is formulated. Conditions are given that allow tumor eradication under constant infusion therapies. Then the optimal scheduling of a vessel disruptive agent in combination with a cytotoxic drug is considered as an optimal control problem. Both theoretical and numerical results on the structure of optimal controls are presented.     

Optimal control for selected cancer chemotherapy ODE models: a view on the potential of optimal schedules and choice of objective function

In this article, four different mathematical models of chemotherapy from the literature are investigated with respect to optimal control of drug treatment schedules. The various models are based on two different sets of ordinary differential equations and contain either chemotherapy, immunotherapy, anti-angiogenic therapy or combinations of these. Optimal control problem formulations based on these models are proposed, discussed and compared. For different parameter sets, scenarios, and objective functions optimal control problems are solved numerically with Bock’s direct multiple shooting method.In particular, we show that an optimally controlled therapy can be the reason for the difference between a growing and a totally vanishing tumor in comparison to standard treatment schemes and untreated or wrongly treated tumors. Furthermore, we compare different objective functions. Eventually, we propose an optimization-driven indicator for the potential gain of optimal controls. Based on this indicator, we show that there is a high potential for optimization of chemotherapy schedules, although the currently available models are not yet appropriate for transferring the optimal therapies into medical practice due to patient-, cancer-, and therapy-specific components.     

宏观生物经济系统线性二次型问题的稳定性分析

本文用控制方法研究宏观生物经济系统,对其进行了线性二次型最优控制设计,给出了该系统的线性二次型问题模型,并对其稳定性进行分析,利用李雅普诺夫稳定性定理,证明了该控制设计的最优控制系统是大范围稳定的,从而得到了该系统在线性二次型控制下保持生物产业经济稳定增长的结论,最后通过仿真验证了结论的正确性．     

Optimal Control of Gypsy Moth Populations

This study investigates an optimal strategy for the cost effective control of gypsy moth populations. Gypsy moth populations cycle between low sparse numbers to high outbreak levels and it is during the outbreak levels that the moths cause extensive damage to plant foliage which can lead to deforestation. Deforestation can result in significant economic damage to infested areas, and consequently, there have been many efforts to control moth populations. One effective method of control is the use of the biocontrol agent, Gypchek, but its production is costly. We develop a mathematical model which combines population dynamics and optimal control of the moth population to explore strategies by which the total cost of the gypsy moth problem (economic damage and cost of Gypchek) can be minimized.     

一类新Willis环脑动脉瘤系统的构造及最优控制

本文基于6-氨基乙酸减小血流速度及缓冲血流波动的性质,在6-氨基乙酸的作用下,构建了一个新的脑动脉瘤数学模型.该模型很好地体现了6-氨基乙酸、血流速度及血流速度变化率三者之间的相互作用关系.鉴于脑动脉瘤的医疗费用颇高及破裂后的死亡率较高,从而有必要从数学的角度研究该模型的最优控制,以致在一定条件下花费最小且医疗效果最佳.本文首先证明了该模型最优控制的存在性;其次通过构造Lagrangian函数及运用最大值原理,证明了最优控制的唯一性.从理论上得到Willis环脑动脉瘤内血流波动最小的条件,这为预防脑动脉瘤的破裂提供了理论依据..     

Combined harvesting of a stage structured prey–predator model incorporating cannibalism in competitive environment

In this paper, we propose a prey–predator system with stage structure for predator. The proposed system incorporates cannibalism for predator populations in a competitive environment. The combined fishing effort is considered as control used to harvest the populations. The steady states of the system are determined and the dynamical behavior of the system is discussed. Local stability of the system is analyzed and sufficient conditions are derived for the global stability of the system at the positive equilibrium point. The existence of the Hopf bifurcation phenomenon is examined at the positive equilibrium point of the proposed system. We consider harvesting effort as a control parameter and subsequently, characterize the optimal control parameter in order to formulate the optimal control problem under the dynamic framework towards optimal utilization of the resource. Moreover, the optimal system is solved numerically to investigate the sustainability of the ecosystem using an iterative method with a Runge–Kutta fourth-order scheme. Simulation results show that the optimal control scheme can achieve sustainable ecosystem. Results are analyzed with the help of graphical illustrations.     

利用泽兰实蝇控制紫茎泽兰的生防策略研究

本文根据野外泽兰实蝇种群15个世代的调查资料,利用契贝谢夫正交多项式拟合了泽兰实蝇、紫茎泽兰的空间格局。阐明了泽兰实蝇空间格局的序列变化及紫茎泽兰空间格局的特点;揭示了泽兰实蝇空间格局的特点受当地主风及寄主紫茎泽兰空间格局特点的影响;并从最优控制系统的角度对泽兰实蝇-紫茎泽兰系统作了初步探讨。首次提出了最佳释放虫量指标为每条虫占有10条枝条。这些结果为多点释放及定点多次释放泽兰实蝇防治紫茎泽兰这一生防策略提供了理论基础。     

Optimal and suboptimal protocols for a class of mathematical models of tumor anti-angiogenesis

Tumor anti-angiogenesis is a cancer treatment approach that aims at preventing the primary tumor from developing its own vascular network needed for further growth. In this paper the problem of how to schedule an a priori given amount of angiogenic inhibitors in order to minimize the tumor volume is considered for three related mathematical formulations of a biologically validated model developed by Hahnfeldt et al. [1999. Tumor development under angiogenic signalling: a dynamical theory of tumor growth, treatment response, and postvascular dormancy. Cancer Res. 59, 4770-4775]. Easily implementable piecewise constant protocols are compared with the mathematically optimal solutions. It is shown that a constant dosage protocol with rate given by the averaged optimal control is an excellent suboptimal protocol for the original model that achieves tumor values that lie within 1% of the theoretically optimal values. It is also observed that the averaged optimal dose is decreasing as a function of the initial tumor volume.     

Revisiting excitation pattern design for magnetic resonance imaging through optimisation of the signal contrast efficiency

The design of excitation signals for Magnetic Resonance Imaging (MRI) is cast as an optimal control problem. Here, we demonstrate that signals other than pulse excitations, which are ubiquitous in MRI, can provide adequate excitation, thus challenging the optimality and ubiquity of pulsed signals. A class of on-resonance piecewise continuous amplitude modulated signals is introduced. It is shown that despite the bilinear nature of the Bloch equations, the spins system response is largely analytically tractable for this class of signals, using Galerkin approximation methods. To challenge the optimality of the pulse excitation, an appropriate cost criterion, the Signal Contrast Efficiency (SCE), is developed. It is to be optimised subject to dynamics expressed by the Bloch equations. To solve the problem the Bloch equation is transferred to the excitation dependent rotating frame of reference. The numerical solutions to the problem for different tissue types show that for a short period of time, pulse excitations provide the maximum signal contrast. However, the problem should be solved for longer periods of time which may result in a different answer than a pulse. For this purpose, the approximate analytic solution which is derived based on averaging the Bloch equation in the excitation dependent rotating frame of reference will be used to find the optimal excitation pattern. The solution to the optimisation problem is potentially useful for all forms of MRI including structural and functional imaging. The objective of this paper is to show that while classically transient response of pulses have been monitored so far, the optimal excitation pattern may be the steady state response of a non-pulse excitation.     

污染环境中可再生资源的最优收获问题

探讨了污染环境下双种群的最优收获问题．利用Pontryagin极大值原理得到一种最优分配方案——处理资源种群体内毒素的努力度与收获资源种群的努力度的分配方案,使经营者的经济收入达到最大,同时也得到次最优均衡解。     

Optimal control strategy of malaria vector using genetically modified mosquitoes

The development of transgenic mosquitoes that are resistant to diseases may provide a new and effective weapon of diseases control. Such an approach relies on transgenic mosquitoes being able to survive and compete with wild-type populations. These transgenic mosquitoes carry a specific code that inhibits the plasmodium evolution in its organism. It is said that this characteristic is hereditary and consequently the disease fades away after some time. Once transgenic mosquitoes are released, interactions between the two populations and inter-specific mating between the two types of mosquitoes take place. We present a mathematical model that considers the generation overlapping and variable environment factors. Based on this continuous model, the malaria vector control is formulated and solved as an optimal control problem, indicating how genetically modified mosquitoes should be introduced in the environment. Numerical simulations show the effectiveness of the proposed control.     

Distributed parameters deterministic model for treatment of brain tumors using Galerkin finite element method

In this paper, we present a distributed parameters deterministic model for treatment of brain tumors using Galerkin finite element method. The dynamic model comprises system of three coupled reaction-diffusion models, involving the tumor cells, the normal tissues and the drug concentration. An optimal control problem is formulated with the goal of minimizing the tumor cell density and reducing the side effects of the drug. A distributed parameters method based on the application of variational calculus is used on an integral-Hamiltonian, which is then used to obtain an optimal coupled system of forward state equations and backward co-state equations. The Galerkin finite element method is used to realistically represent the brain structure as well as to facilitate computation. Finally a three-dimensional test case is considered and partitioned into a set of spherical finite elements, using tri-linear basis functions, except for the elements affected by singularities of polar and azimuthal angles, as well as the origin.