A discrete method of optimal control based upon the cell state space concept

A discrete method of optimal control is proposed in this paper. The continuum state space of a system is discretized into a cell state space, and the cost function is discretized in a similar manner. Assuming intervalwise constant controls and using a finite set of admissible control levels (u) and a finite set of admissible time intervals (), the motion of the system under all possible interval controls (u, ) can then be expressed in terms of a family of cell-to-cell mappings. The proposed method extracts the optimal control results from these mappings by a systematic search, culminating in the construction of a discrete optimal control table.The possibility of expressing the optimal control results in the form of a control table seems to give this method a means to make systems real-time controllable.Dedicated to G. LeitmannThe material is based upon work supported by the National Science Foundation under Grant No. MEA-82-17471. The author is also indebted to Professor G. Leitmann for his many helpful comments.     

Computation of suboptimal randomized strategies for steering the random motion of a point under partial observation

Consider the random motion in the plane of a pointM, whose velocityv=(v ₁,v ₂) is perturbed by an ²-valued Gaussian white noise. Only noisy nonlinear observations taken on the point location (state) are available toM. The velocityv is of the formv(y)= _u (u ₁,u ₂) _y (du), wherey denotes the value of the observed signal,U is the range of the velocity, and, for eachy, _y is a probability measure on (U). Using the available observations, the pointM wishes to steer itself into a given target set by choosing a randomized strategy ={ _y :y ²}. Sufficient conditions on weak optimal randomized strategies are derived. An algorithm for computing weak suboptimal randomized strategies is suggested, and the strategies are computed for a variety of cases.This work was partially supported by a grant from Control Data.     

Optimal dynamic advertising policies for hereditary processes

In this paper, the Nerlove-Arrow model of optimal dynamic advertising policies is generalized by assuming a general probability distribution of the forgetting time, rather than the exponential one. A control problem with integrodifferential equations of motion is defined for which the transitory and steady-state properties of the optimal advertising policy are examined. The effects of assumptions like IHR-distributions and DHR-distributions, the existence of an upper bound for the forgetting time, etc., are explained. It is shown that there are two (in the case of an exponential distribution even three) different current-value adjoint functions associated with the problem, and relations between the two (three) are established. Also provided is a sensitivity analysis.Thanks are due to G. Feichtinger and S. Jorgensen for useful discussions.     

The existence of sensitive optimal policies in two multi-dimensional queueing models

Recently Dekker and Hordijk [3,4] introduced conditions for the existence of deterministic Blackwell optimal policies in denumerable Markov decision chains with unbounded rewards. These conditions include- uniform geometric recurrence.The-uniform geometric recurrence property also implies the existence of average optimal policies, a solution to the average optimality equation with explicit formula's and convergence of the value iteration algorithm for average rewards. For this reason, the verification of-uniform geometric convergence is also useful in cases where average and-discounted rewards are considered.On the other hand,-uniform geometric recurrence is a heavy condition on the Markov decision chain structure for negative dynamic programming problems. The verification of-uniform geometric recurrence for the Markov chain induced by some deterministic policy together with results by Sennott [14] yields the existence of a deterministic policy that minimizes the expected average cost for non-negative immediate cost functions.In this paper-uniform geometric recurrence will be proved for two queueing models: theK competing queues and the two centre open Jackson network with control of the service rates.The research of the author is supported by the Netherlands Organization for Scientific Research N.W.O.     

A review of statistical models for global optimization

A review of statistical models for global optimization is presented. Rationality of the search for a global minimum is formulated axiomatically and the features of the corresponding algorithm are derived from the axioms. Furthermore the results of some applications of the proposed algorithm are presented and the perspectives of the approach are discussed.     

Numerical solution of minimax problems of optimal control,part 1

This paper contains general transformation techniques useful to convert minimax problems of optimal control into the Mayer-Bolza problem of the calculus of variations [Problem (P)]. We consider two types of minimax problems: minimax problems of Type (Q), in which the minimax function depends on the state and does not depend on the control; and minimax problems of Type (R), in which the minimax function depends on both the state and the control. Both Problem (Q) and Problem (R) can be reduced to Problem (P).For Problem (Q), we exploit the analogy with a bounded-state problem in combination with a transformation of the Jacobson type. This requires the proper augmentation of the state vectorx(t), the control vectoru(t), and the parameter vector , as well as the proper augmentation of the constraining relations. As a result of the transformation, the unknown minimax value of the performance index becomes a component of the parameter vector being optimized.For Problem (R), we exploit the analogy with a bounded-control problem in combination with a transformation of the Valentine type. This requires the proper augmentation of the control vectoru(t) and the parameter vector , as well as the proper augmentation of the constraining relations. As a result of the transformation, the unknown minimax value of the performance index becomes a component of the parameter vector being optimized.In a subsequent paper (Part 2), the transformation techniques presented here are employed in conjunction with the sequential gradient-restoration algorithm for solving optimal control problems on a digital computer; both the single-subarc approach and the multiple-subarc approach are discussed.This research was supported by the National Science Foundation, Grant No. ENG-79-18667, and by Wright-Patterson Air Force Base, Contract No. F33615-80-C3000. This paper is a condensation of the investigations reported in Refs. 1–7. The authors are indebted to E. M. Coker and E. M. Sims for analytical and computational assistance.     

Facility location in the presence of forbidden regions,I: Formulation and the case of Euclidean distance with one forbidden circle

A constrained form of the Weber problem is formulated in which no path is permitted to enter a prespecified forbidden region $\text{R}$ of the plane. Using the calculus of variations the shortest path between two points $\text{x}\text{,}\text{y}\text{?}\text{R}$ which does not intersect $\text{R}$ is determined. If $\text{d(}\text{x}\text{,}\text{y}\text{)}$ is unconstrained distance, we denote the shortes distance along a feasible path by $\text{d}\text{(}\text{x}\text{y}\text{)}$. The constrained Weber problem is, then: given points $\text{x}{}_{\mathrm{j}}\text{?}\text{R}$ and positive weights w_j, j = 1,2,…,n, find a point $\text{x}\text{?}\text{R}$ such that

$\text{f(}\text{x}\text{)=}\text{Σ}\text{n}\text{j=1}\text{d(}\text{x}\text{,}\text{x}{}_{\mathrm{j}}\text{)}$

is a minimum.An algorithm is formulated for the solution of this problem when $\text{d(}\text{x}\text{,}\text{y}\text{)}$ is Euclidean distance and $\text{R}$ is a single circular region. Numerical results are presented.     

含一般模糊弹性约束的广义模糊变量线性规划

建立并讨论了一类含有一般模糊弹性约束的广义模糊变量线性规划问题.首先,简单介绍了结构元方法并对结构元加权排序中权函数表征决策者风险态度进行了深入分析.然后选取风险中性的决策者来定义序关系,应用Verdegay模糊线性规划方法将含一般模糊弹性约束的广义模糊变量线性规划转化经典的线性规划问题,简化了原问题的求解.最后通过数值算例进一步说明了该方法的有效性.     

Determination/estimation of an optimal replacement interval under minimal repair

In this paper the problem of estimation of an optimal replacement interval for a system which is minimally repaired at failures is studied. The problem is investigated both under a parametric and a nonparametric form of the failure intensity of the system. It is assumed that observational data from n systems are available. Some asymptotic results are shown. A graphical procedure for determining/estimating an optimal replacement interval is presented. The procedure is particularly valuable for sensitivity analyses, for example with respect to the costs involved.     

Difference systems of sets based on cosets partitions

Difference systems of sets （DSS） are combinatorial configurations that arise in connection with code synchronization. This paper proposes a new method to construct DSSs, which uses known DSSs to partition some of the cosets of Zv relative to subgroup of order k, where v = km is a composite number. As applications, we obtain some new optimal DSSs.