Optimal Evasion from a Pursuer with Delayed Information

A class of prescribed duration pursuit–evasion problems with first-order acceleration dynamics and bounded controls is considered. In this class, the pursuer has delayed information on the lateral acceleration of the evader, but knows perfectly the other state variables. Moreover, the pursuer applies a strategy derived from the perfect information pursuit–evasion game solution. Assuming that the evader has perfect information on all the state variables as well as on the delay of the pursuer and its strategy, an optimal evasion problem is formulated. The necessary optimality conditions indicate that the evader optimal control has a bang–bang structure. Based on this result, two particular cases of the pursuer strategy (continuous and piecewise continuous in the state variables) are considered for the solution of the optimal evasion problem. In the case of the continuous pursuer strategy, the switch point of the optimal control can be obtained as a root of the switch function. However, in the case of the piecewise continuous (bang–bang) pursuer strategy, this method fails, because of the discontinuity of the switch function at this very point. In this case, a direct method for obtaining the switch point, based on the structure of the solution, is proposed. Numerical results illustrating the theoretical analysis are presented leading to a comparison of the two cases.     

Optimal life strategies in organisms exposed to recurrent critical events

The optimal partition of energy between survival and reproduction is considered for a population subject to recurrent and potentially lethal critical events. The best strategy is found by maximizing fitness, a functional derived from the Lotka equation. The dynamics is governed by a second-order, age-varying, nonlinear system. The energy storage and the probability of survival are the state variables, while the amounts of energy placed into and withdrawn from the storage are the controls. The optimal life strategy is shown to be as follows: build up the storage at the very beginning of life, and then progressively deplete it to resist the critical events.This work was partially supported by MURST Project Ricerche sui Fondamenti della Conservazione della Natura. The authors would like to thank C. Ricci for suggesting the problem and C. Matessi for helpful discussion.     

Singular fuel-optimal space trajectories based on linearization about a point in circular orbit

Singular solutions of a linearized problem of determining fuel-optimal trajectories of a spacecraft with bounded thrust near a satellite in circular orbit are investigated in detail. Mathematical results are presented which classify and discuss degeneracy of the singular solutions and which show that strictly singular and nonsingular solutions are mutually exclusive for fixed-end conditions. These results are consistent with those found using other approaches for other linear models, but they are not characteristic of the type of results obtained without linearization.     

Optimal economic growth with foreign trade and a learning industry

Ryder (Ref. 1) has developed a simple two-sector macroeconomic model incorporatinglearning by doing as a determinant of the growth of productivity-enhancing experience and its effect on foreign trade. In this paper, optimal foreign trade and capital allocation policies are determined, in the context of the above model, for ranges of the international trade price not considered by Ryder. An extension of Ryder's model to include a dual trade price structure is briefly discussed. A specific numerical example is used to ascertain the configuration of the various features occurring in the extremal fields at different price levels.     

Singular Arcs in the Generalized Goddard’s Problem

We investigate variants of Goddard’s problems for nonvertical trajectories. The control is the thrust force, and the objective is to maximize a certain final cost, typically, the final mass. In this article, performing an analysis based on the Pontryagin maximum principle, we prove that optimal trajectories may involve singular arcs (along which the norm of the thrust is neither zero nor maximal), that are computed and characterized. Numerical simulations are carried out, both with direct and indirect methods, demonstrating the relevance of taking into account singular arcs in the control strategy. The indirect method that we use is based on our previous theoretical analysis and consists in combining a shooting method with an homotopic method. The homotopic approach leads to a quadratic regularization of the problem and is a way to tackle the problem of nonsmoothness of the optimal control. Support from the French Space Agency CNES (Centre National d’Etudes Spatial) is gratefully acknowledged.     

Proportional navigation and the game of two cars

A stochastic version of Isaacs's game of two cars is considered. The motion of the players is confined to the pursuer's effective operation zoneD _P, and the cost function of the game is the probability of the event: {Before the evader enters his safe zone, the evader enters the pursuer's killing zoneK _P, at somet, 0tT, or the evader stays at the domainD _P–K _P, for allt[0,t ₀],t ₀>T}. By numerically solving a nonlinear parabolic boundary-value problem on a generalized torus in ³, it is shown that, for a range of values of some parameter, a proportional navigation guidance law is an optimal feedback pursuit strategy.     

Optimal investment in the reclamation of eutrophic water bodies

Eutrophication, i.e., the abnormal growth of phytoplankton, is considered in this note, which focuses on the optimal treatment of eutrophic water bodies. The issue is addressed by the use of a nonlinear model where phytoplankton and the number of wastewater treatment plants in operation are the state variables. The decision maker is a governmental agency which has to define the time pattern of investment in new plants so as to minimize the present value of environmental and treatment costs. The optimal solution is shown to have the following features. First, the optimal size for the wastewater treatment system is attained in minimum time. Subsequently, investment replaces wornout treatment plants, and phytoplankton adjusts asymptotically to its optimal equilibrium value.This work was supported by Centro Teoria dei Sistemi—CNR, Milano and by Fondazione ENI E. Mattei.     

Necessary Condition for a Strong Relative Minimum for Singular Optimal Control

A necessary condition is developed for a strong relative minimum for a first-order singular optimal control. It is shown that the strong condition is identical with the existing weak condition, namely, the generalized Legendre-Clebsch condition. It is conjectured that the strong condition for a second-order singular control will be the same as the corresponding weak condition.     

Simple-motion pursuit-evasion differential games,part 2: Optimal evasion from proportional navigation guidance in the deterministic and stochastic cases

The problem of the optimal evasion from a pursuer employing proportional navigation guidance is considered. Thus, we consider the problem of maximizing the time-to-capture when the problem parameters, namely, the speed ratio and the proportional navigation constant, are such that capturability of the evader is guaranteed.     

双重委托-代理理论在税收流失中的应用

本文研究了我国国税局和地税局在税收征管上的合作问题。首次应用双重委托一代理理论对该问题进行了经济分析。研究发现,国税局和地税局在税收征收和税收监管上完全合作时,相对而言,能最大化它们的总效用;而当国税局和地税局只在税收监管上合作时,税收征收上的“正外部性”将导致相对“过剩”的监管;最后,如果国税局和地税局完全不合作时,监管上的“搭便车”效应是影响博弈结果的主导因素。研究结果有一定的现实意义。     

Computing Estimates in the Proportional Odds Model

The semiparametric proportional odds model for survival data is useful when mortality rates of different groups converge over time. However, fitting the model by maximum likelihood proves computationally cumbersome for large datasets because the number of parameters exceeds the number of uncensored observations. We present here an alternative to the standard Newton-Raphson method of maximum likelihood estimation. Our algorithm, an example of a minorization-maximization (MM) algorithm, is guaranteed to converge to the maximum likelihood estimate whenever it exists. For large problems, both the algorithm and its quasi-Newton accelerated counterpart outperform Newton-Raphson by more than two orders of magnitude.     

Optimal Value and Post Optimal Solution in a Transportation Problem

In this work, we analyse the transportation problem of a real-life situation by obtaining the optimal feasible solutions, thus carrying out the sensitivity analysis of the problem. The work utilises the data obtained from the Asejire and Ikeja plants of Coca-Cola company, aiming to aid decision-making regarding the best possible options to satisfy customers at the barest minimum cost of transportation. Rerunning the optimization of a problem is an expensive scheme for gathering and obtaining enough data required for a problem. Thus, to minimize the transportation cost, the sensitivity analysis of parameters is a good tool to determine the behaviour of some input parameters where the values of these parameters are varied arbitrarily such that optimal results are verified. Maple 18 Software is used to solve the problem and the result obtained is compared with the values evaluated from northwest corner method, least cost method and Vogel''s approximation method. The study critically shows how a little change in a unit or more of any model parameter affects the expected results.     

On Necessary Optimality Conditions for Singular Controls in Dynamical Systems with a Delay in Control

Abstract

In this article, an optimal control problem with a delay in control is considered. The second-order necessary condition is obtained for the optimality of singular (in the sense of the maximum principle) control. Also, the notion of degenerate singular control of order k (k?≥?1) is introduced and for optimality of this, the high-order necessary condition is obtained. Moreover, while studying the problem, one of the strengthened version of an analog of the maximum principle is shown. Finally, the rich content of the obtained results is illustrated by specific examples.     

Some Properties of the Optimal Preconditioner and the Generalized Superoptimal Preconditioner

<正>The optimal preconditioner and the superoptimal preconditioner were proposed in 1988 and 1992 respectively.They have been studied widely since then.Recently, Chen and Jin[6]extend the superoptimal preconditioner to a more general case by using the Moore-Penrose inverse.In this paper,we further study some useful properties of the optimal and the generalized superoptimal preconditioners.Several existing results are extended and new properties are developed.     

On the Eigenvalues and Eigenfunctions of the Sturm--Liouville Operator with a Singular Potential

In this paper we consider the Sturm--Liouville operators generated by the differential expression -y+q(x)y and by Dirichlet boundary conditions on the closed interval [0,]. Here q(x) is a distribution of first order,, i.e., q(x)dx L ₂[0,]. Asymptotic formulas for the eigenvalues and eigenfunctions of such operators which depend on the smoothness degree of q(x) are obtained.     

Singular Set of a Convex Potential in Two Dimensions

Let u be a convex potential of the optimal transfer map from a convex open set X to a nonconvex open set Y in the plane. If u only has singularities whose sets of supports are one dimension, then under some mild assumptions on Y, we show that the singular set of u are disjoint union of countably many C ¹ curves. Or we can say that the singular set is a C ¹ manifold.     

Beyond all orders: Singular perturbations in a mapping

Summary We consider a family ofq-dimensional (q>1), volume-preserving maps depending on a small parameterε. Asε → 0⁺ these maps asymptote to flows which attain a heteroclinic connection. We show that for smallε the heteroclinic connection breaks up and that the splitting between its components scales withε likeε ^γexp[-β/ε]. We estimateβ using the singularities of theε → 0⁺ heteroclinic orbit in the complex plane. We then estimateγ using linearization about orbits in the complex plane. These estimates, as well as the assertions regarding the behavior of the functions in the complex plane, are supported by our numerical calculations. Deceased.     

在空间L［a，b］上收敛的几个奇异积分的构造特征

本文构造了一类奇异积分，并讨论它们在Ｌ［ａ，ｂ］上的收敛性．从结构上统一处理了三个重要的奇异积分。     

Singular surfaces in a linear pursuit-evasion game with elliptical vectograms

In a linear pursuit-evasion game with elliptical vectograms, singular surfaces appear in the plane of symmetry containing the ellipses minor axes and, as a consequence, locally nonsmooth isocost tubes are generated. Both dispersal and attractive focal surfaces are encountered. The dispersal surface is a zone of initial conditions for trajectories leaving the plane of symmetry. Optimal trajectories attracted by the focal surface merge tangentially to the plane of symmetry and remain there until the boundary of the dispersal zone is reached. Determination of the saddle-point strategies in the focal surface leads to constructing the isocost surfaces in the entire game space.