Solutions for a class of optimal control problems with time delay,part 2

In this work, we analyze the structure of optimal solutions for a class of optimal control problems with time delay whose state is governed by a nonlinear autonomous Volterra integrodifferential equation. We establish that an almost optimal finite trajectory defined on an interval [0, ] is contained in a small neighborhood of the optimal steady state for all , where is a constant which depends on the neighborhood and does not depend on .The author thanks A. Leizarowitz for fruitful discussions.     

Optimal natural dualities. II: General theory

A general theory of optimal natural dualities is presented, built on the test algebra technique introduced in an earlier paper. Given that a set of finitary algebraic relations yields a duality on a class of algebras , those subsets of which yield optimal dualities are characterised. Further, the manner in which the relations in are constructed from those in is revealed in the important special case that generates a congruence-distributive variety and is such that each of its subalgebras is subdirectly irreducible. These results are obtained by studying a certain algebraic closure operator, called entailment, definable on any set of algebraic relations on . Applied, by way of illustration, to the variety of Kleene algebras and to the proper subvarieties of pseudocomplemented distributive lattices, the theory improves upon and illuminates previous results.

     

Controlled and optimally controlled multiplexing systems: A numerical exploration

Large controlled multiplexing systems are approximated by diffusion type processes yielding a very efficient way of approximation and good numerical methods. The limit equations are an efficient aggregation of the original system, and provide the basis of the actual numerical approximation to the control problem. The numerical approximations have the structure of the original problem, but are generally much simpler. The control can occur in a variety of places; e.g., leaky bucket controllers, control of marked cells at the transmitter buffer, or control of the transmitter speed. From the point of view of the limit equations, those are equivalent. Various forms of the optimal control problem are explored, where the main aim is to control or balance the losses at the control with those due to buffer overflow. It is shown that much can be saved via the use of optimal controls or reasonable approximations to them. We discuss systems with one to three classes of sources, various aggregation methods and control approximation schemes. There are qualitative comparisons of various systems with and without control and a discussion of the variations of control and performance as the systems data and control bounds vary. The approach is a very useful tool for providing both qualitative and quantitative information which would be hard to get otherwise. The results have applications to various forms of the ATM and broadband integrated data networks.The work was partially supported by AFOSR-91-0375 and (AFOSR) F49620-92-J-088-1DEF.The work was partially supported by grants (AFOSR) F49620-92-J-008-1DEF, AFOSR-91-03750.This work was partially supported by DAAH04-93-0070 (ARO) and AFOSR-91-0375.     

A full information pricing problem for the sale of several identical commodities

Summary k commodities of the same kind are for sale in a finite time intervalI, and potential customers arrive according to a Poisson process. It is assumed that no customer may buy more than one unit. The seller has to fix, for eacht I, a price for which he is willing to sell a commodity at timet. This decision is based on the knowledge of the probabilities that a customer appearing at timet will accept a price of at mostx monetary units (for allt I andx > 0). There is no possibility of recall of previous customers. We characterize an optimal price function and the maximal expected gain. Further the relation to the persistency problem of Elfving (1967) is exhibited.

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Zusammenfassung k gleichartige Güter sollen in einem endlichen ZeitintervallI an Kunden verkauft werden, deren Ankunftszeitpunkte einen Poisson-Prozeß bilden. Es wird angenommen, daß kein Kunde mehr als eine Einheit kauft. Der Verkäufer muß für jedest I einen Preis angeben, zu dem er eine Einheit zum Zeitpunktt im Falle einer Nachfrage verkaufen würde. Es wird vorausgesetzt, daß er für jedest I und jedesx>0 die Wahrscheinlichkeit kennt, daß ein zur Zeitt eintreffender Kunde höchstens den Preisx für das Gut akzeptiert; die Möglichkeit, auf frühere Kunden zurückzugreifen, ist ausgeschlossen. Für diese Situation wird eine optimale Preisfunktion sowie der zugehörige maximale erwartete Erlös bestimmt. Ferner wird die Beziehung dieses Modells zu einem verwandten Problem von Elfving (1967) herausgestellt.

     

Optimum conditions for transformed Panax ginseng hairy roots in flask culture

Panax ginseng hairy roots were transformed by Agrobacterium rhizogenes KTCT 2744. They showed an active branching pattern and fast growth in hormone-free medium, and good growth at 23°C, pH 5.8, 1/2 MS medium, and 3% sucrose. Sucrose provided the highest growth among seven carbon sources tested. Six complex media were also tested. In the combined sugar study, hairy roots grew better on sucrose without glucose or fructose than with glucose or fructose. In the 1/2 MS basal medium, 30 mM in nitrogen and 0.62 mM phosphate salt concentration was the optimum. The growth ratio was maximal at an inoculum size of 0.4% (w/v). Crude saponin and polysaccharide levels were also measured.     

Optimal scheduling of reservoir releases during flood: Deterministic optimization problem,part 1, procedure

In this paper, we are concerned with the optimal scheduling of water releases from retention reservoirs during flood, with the objective of minimizing flood damages at the important damage centers downstream of the reservoirs. Unlike in most other papers devoted to this subject, the flood routing equations are nonlinear. The performance index of the problem leads to a minimax optimal control problem. For this problem, the necessary optimality conditions are provided and a version of the feasible directions method is proposed.The research reported here has been supported by the Central Basic Research Program CPBP-03.09, Metody Analizy i Uytkowania Zasobow Wodnych, Polish Academy of Sciences, Warsaw, Poland. This support is kindly acknowledged.     

Testing problems with nuisance parameters: Linear models under non-classical assumptions

Summary Most testing problems involve a multidimensional parameter, only one component of which characterizes the hypotheses. In a mathematically strong sense, there are only two methods to cope with the remaining component, the so called nuisance parameter. Firstly, conditioning on a sufficient and complete statistic and, secondly, reduction by invariance. Both methods require strong assumptions on the underlying class of distributions. Therefore the local asymptotic approach is reviewed, which extends Neyman's theory ofC()-tests and yields, in a unified way, approximate optimal test statistics for smoothly indexed families. That device is applied to non-normal heteroscedastic linear models.This is a written account of an invited lecture delivered by the second author on occasion of the 12. Symposium über Operations Research, Passau, 9.–11. 9. 1987.     

Modified quasilinearization algorithm for optimal control problems with nondifferential constraints and general boundary conditions

This paper considers the numerical solution of the problem of minimizing a functionalI, subject to differential constraints, nondifferential constraints, and general boundary conditions. It consists of finding the statex(t), the controlu(t), and the parameter so that the functionalI is minimized while the constraints are satisfied to a predetermined accuracy.The modified quasilinearization algorithm (MQA) is extended, so that it can be applied to the solution of optimal control problems with general boundary conditions, where the state is not explicitly given at the initial point.The algorithm presented here preserves the MQA descent property on the cumulative error. This error consists of the error in the optimality conditions and the error in the constraints.Three numerical examples are presented in order to illustrate the performance of the algorithm. The numerical results are discussed to show the feasibility as well as the convergence characteristics of the algorithm.This work was supported by the Electrical Research Institute of Mexico and by CONACYT, Consejo Nacional de Ciencia y Tecnologia, Mexico City, Mexico.     

Parameter-dependent renewal theorems with applications

Let ₁, ₂, ... be a sequence of i.i.d. random variables with positive mean and finite variance and letr(b), b0, be real numbers tending to 0 asb . Definings _n=₁+...+_n andS _n=S_n(b)=s_n+r(b)n, the stopping time =(b)=inf {n>/1:S_n >b} whereb=b(b) , will be considered with special regard to the excess over the boundaryR _b=s+r(b)–b. It turns out that the limiting distribution ofR _b is the same as in the caser(b)0 for allb. Proving this, Blackwell's renewal theorem and its integral version have to be established first in the above stated situation. Finally, an expansion ofE to vanishing terms asb will be provided and applied to some examples arising in economics.

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Zusammenfassung Seien ₁, ₂, ... unabhängige identisch verteilte Zufallsgrößen mit positivem Erwartungswert und endlicher Varianz sowier(b), b0, reelle Zahlen mitr(b)0 für b. Sei ferners ₁, s₂, ... der zugehörige Summenprozeß,S _n= S_n(b)=s_n+r(b)n fürn1 und =(b)=inf {n1: S_n>b, wobeib=b(b) fürb . Es wird gezeigt, daß die asymptotische Verteilung des ExzessesR _b=s +r(b) –b mit der im Fallr(·)0 übereinstimmt. Dazu werden sowohl das Blackwellsche Erneuerungstheorem als auch seine Integralversion in der vorher beschriebenen parameterabhängigen Situation geeignet formuliert und bewiesen. Als Folgerung ergibt sich dann eine asymptotische Entwicklung vonE(b) fürb bis zu Termen o(1). Anh- and einiger Beispiele aus dem ökonomischen Bereich wird schließlich noch aufgezeigt, wo Approximationen fürE(b) von Interesse sein können.