具有年龄结构的种群线性动力系统的最优收获控制问题

§ 1　Introduction and setting of the problemThe optimal control of age-dependent population dynamics has been intensivelystudied in the last two decades and there is now a vast stock of literature on the topic ofoptimal control problems ofage-structured population dynamics.(see [1 -9] ) .To the bestof our knowledge,the works of Brokate[3,4] are the firstto deal with this topic.Since then,many authors devote to the optimal harvesting problem.In this aspect,we refere to thefundamental papers o…     

A variational method of optimal control problems for nonlinear Schrödinger equations

Calculating the control function that is in the coefficients of the nonlinear Schrodinger equation and minimizes the objective functional consisting of two integrals is main aim of this article. After proving the existence and uniqueness of the solution, we have constituted an algorithm using the method of gradient of projection. Results are tested on a quantum‐mechanical problem. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 82–89, 2004.     

An inverse optimal problem in discrete-time stochastic control

In this paper, we study an inverse optimal problem in discrete-time stochastic control. We give necessary and sufficient conditions for a solution to a system of stochastic difference equations to be the solution of a certain optimal control problem. Our results extend to the stochastic case the work of Dechert. In particular, we present a stochastic version of an important principle in welfare economics.     

Optimal control of age-structured population dynamics for spread of universally fatal diseases

This article is concerned with the optimal control problem of age-structured population dynamics for the spread of universally fatal diseases. The existence and uniqueness of solution of the system, which consists of a group of partial differential equations with nonlocal boundary conditions, is proved. The Dubovitskii and Milyutin functional analytical approach is adopted in the investigation of the Pontryagin maximum principle of the system. The necessary condition is presented for the optimal control problem in fixed final horizon case. Finally, a remark on how to utilize the obtained results is also made.     

Numerical analysis for the approximation of optimal control problems with pointwise observations

In this paper, we study the numerical methods for optimal control problems governed by elliptic PDEs with pointwise observations of the state. The first order optimality conditions as well as regularities of the solutions are derived. The optimal control and adjoint state have low regularities due to the pointwise observations. For the finite dimensional approximation, we use the standard conforming piecewise linear finite elements to approximate the state and adjoint state variables, whereas variational discretization is applied to the discretization of the control. A priori and a posteriori error estimates for the optimal control, the state and adjoint state are obtained. Numerical experiments are also provided to confirm our theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.     

Optimal control of age-structured population dynamics for spread of universally fatal diseases II

In our preceding paper [Sun B, Wu MX. Appl. Anal. 2013;92:901–921], we investigated an optimal control problem of age-structured population dynamics for spread of universally fatal diseases and derived the necessary optimality condition for the problem in fixed final horizon case. As a follow-up, in this paper, under weaker additional conditions, we address ourselves to the investigation of the foregoing system in free final horizon case and present further new results of current interests.     

Numerical Simulation of the Two-Hydrodynamic Film Thickness

We investigate, from a numerical point of view, the coefficients identification problem, for the Elrod–Adams model of cavitation, in the frame of the hydrodynamic lubrication. We relax the control problem, and propose a relaxed augmented Lagrangian algorithm, with a given loop length. Some numerical results are given.     

Remarks on the algorithm of sakawa for optimal control problems

A general optimal control problem for ordinary differential equations is considered. For this problem, some improvements of the algorithm of Sakawa are discussed. We avoid any convexity assumption and show with an example that the algorithm is even applicable in cases in which no optimal control exists.     

Numerical method for solving an inverse problem for a population model

The inverse problem of determining the growth rate coefficient of biological objects from additional information on their time-dependent density is considered. Two nonlinear integral equations are derived for the unknown coefficient, which is determined on part of its domain from one equation and on the remaining part from the other equation. The nonlinear integral equations are solved by iterative methods. The convergence conditions for the iterative methods are formulated, and results of numerical experiments are presented.     

A robust optimal preconditioner for the mixed finite element discretization of elliptic optimal control problems

In this paper, we consider the efficient solving of the resulting algebraic system for elliptic optimal control problems with mixed finite element discretization. We propose a block‐diagonal preconditioner for the symmetric and indefinite algebraic system solved with minimum residual method, which is proved to be robust and optimal with respect to both the mesh size and the regularization parameter. The block‐diagonal preconditioner is constructed based on an isomorphism between appropriately chosen solution space and its dual for a general control problem with both state and gradient state observations in the objective functional. Numerical experiments confirm the efficiency of our proposed preconditioner.     

On the optimal harvesting of size-structured population dynamics

This work is concerned with a kind of optimal control problem for a size-structured biological population model. Well-posedness of the state system and an adjoint system are proved by means of Banach’s fixed point theorem. Existence and uniqueness of optimal control are shown by functional analytical approach. Optimality conditions describing the optimal strategy are established via tangent and normal cones technique. The results are of the first ones for this novel structure.     

SIR类型新型冠状病毒肺炎多阶段最优控制模型

新型冠状病毒肺炎(COVID-19)疫情在全球范围传播,给人们的健康带来了严重的威胁。面对疫情发展预期数据,我们需要在有限医疗资源的情况下确定疫情传播参数,以指导主要防疫措施的实施力度。本文采用SIR类型的模型描述新冠肺炎疫情发展,并建立多阶段最优控制模型确定疫情传播参数。为了高效确定参数取值,我们建立多项式时间可计算的半定规划近似模型。基于世界卫生组织发布的数据,我们求解近似模型,得到描述给定时段内美国新冠肺炎疫情发展态势的疫情传播参数,并分析疫情防控策略。     

Numerical methods for singular optimal control

Methods are described for the numerical solution of singular optimal control problems. A simple method is given for solving a class of problems which form a transition from nonsingular to singular cases. A procedure is given for determining the structure of a singular problem if it is initially unknown. Several numerical examples are presented.This work is based on the author's PhD Dissertation at The Hatfield Polytechnic, Hatfield, Hertfordshire, England.     

Local discontinuous galerkin approximation of convection‐dominated diffusion optimal control problems with control constraints

In this article, we investigate local discontinuous Galerkin approximation of stationary convection‐dominated diffusion optimal control problems with distributed control constraints. The state variable and adjoint state variable are approximated by piecewise linear polynomials without continuity requirement, whereas the control variable is discretized by variational discretization concept. The discrete first‐order optimality condition is derived. We show that optimization and discretization are commutative for the local discontinuous Galerkin approximation. Because the solutions to convection‐dominated diffusion equations often admit interior or boundary layers, residual type a posteriori error estimate in L² norm is proved, which can be used to guide mesh refinement. Finally, numerical examples are presented to illustrate the theoretical findings. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 339–360, 2014     

Numerical solution and optimal control for fractional tumor immune model

In this article, the numerical model of fractional tumor immunity has been described. We have proved and analyzed the model does have a stable solution. In addition to this, the optimal control of their form as well as the numerical approach for the simulation of the control problem, are both brought up and examined. We have presented evidence that demonstrates the existence of the solution. We use an algorithm modeled after the generalized Adams-Bashforth-Moulton style (GABMS) to solve the fractional tumor immune model. This amendment is predicated on changing the form to a memristive one for the first time because such a notion is being utilized for the first time to control this ailment. The dissection results have been interpreted using numerical simulations we created. To calculate the results, we relied on the Maple 15 programming language.     

Penalty approximation of a Sobolev optimal control problem with state constraints

In this paper, we study the approximation by the penalty method of a control problem governed by a pseudo-parabolic equation with a noncoercive control functional and with control and state constraints. The existence of solutions to the penalized problems is established. In addition, the convergence of the penalized problems to the solution, the Lagrange multipliers, and the minimum value of the original problem is studied. The results apply to Sobolev and parabolic equations as well.This work was partially supported by the National Science Foundation, Grant No. MCS-79-02037. The author would like to thank Professor A. B. Schwarzkopf for his helpful comments on this paper.     

Robust error estimates for the finite element approximation of elliptic optimal control problems

In this paper, we consider the finite element approximation of an elliptic optimal control problem. Based on an assumption on the adjoint state of the continuous problem with a small parameter, which represents a regularization of the bang-bang type control problem, we derive robust a priori error estimates for optimal control and state and a posteriori error estimate is also presented. Numerical experiments confirm our theoretical results.     

A numerical approximation for Volterra’s population growth model with fractional order

This paper presents a numerical scheme for approximate solutions of the fractional Volterra’s model for population growth of a species in a closed system. In fact, the Bessel collocation method is extended by using the time-fractional derivative in the Caputo sense to give solutions for the mentioned model problem. In this extended of the method, a generalization of the Bessel functions of the first kind is used and its matrix form is constructed. And then, the matrix form based on the collocation points is formed for the each term of this model problem. Hence, the method converts the model problem into a system of nonlinear algebraic equations. We give some numerical applications to show efficiency and accuracy of the method. In applications, the reliability of the technique is demonstrated by the error function based on accuracy of the approximate solution.     

A preconditioned MinRes solver for time‐periodic parabolic optimal control problems

This work is devoted to the multiharmonic analysis of parabolic optimal control problems in a time‐periodic setting. In contrast to previous approaches, we include the cases of different control and observation domains, the observation in certain energy spaces, and the presence of control constraints. In all these cases, we propose a new preconditioned MinRes solver for the frequency domain equations and show that this solver is robust with respect to the space and time discretization parameters as well as the involved ‘bad’ model parameters of the state equation. Copyright © 2012 John Wiley & Sons, Ltd.     

Optimal control and optimal trajectories of regional macroeconomic dynamics based on the Pontryagin maximum principle

The Pontryagin maximum principle is used to prove a theorem concerning optimal control in regional macroeconomics. A boundary value problem for optimal trajectories of the state and adjoint variables is formulated, and optimal curves are analyzed. An algorithm is proposed for solving the boundary value problem of optimal control. The performance of the algorithm is demonstrated by computing an optimal control and the corresponding optimal trajectories.