On optimal parameter selection

Pontryagin's results on optimal parameter selection problems [1, p. 193] have been found to contain some errors, as reported in Boltyanskii [2, p. 263] and Hofer and Sagirow [4]. In this note, it is shown that Boltyanskii's results (on optimal parameters) are easily derived from Gamkrelidze's general maximum principle [3], which, in contrast to the former, also yields an explicit solution to the problem in certain special cases.     

相空间坐标受限时的最优控制问题

对于相空间坐标受限的最优控制问题,P.B.加姆克列利德泽等得到了控制最优性的必要条件(即熟知的极大值原理)和衔接点处的“跳跃条件”.但是文献[1]中所用的数学工具不是初等的,推证过程也较繁长,不易为工程技术界所理解.本文在应用文献[1]中某些分析的基础上,采用文献[2,3]中的方法处理这一问题,作法较为简单,而且除了得到[1]中的结论外,还可得到一些新的结果,例如,文中所得到的某些充分条件以及对于线性系统“小范围”最优性的必要充分条件等.     

环Fp+uFp+vFp上的(1+u+v)-循环码

常循环码是一类重要的线性码,由于其纠错性能易于分析,因而广泛应用于实践中,同时,利用有限环上常循环码还可以直接构造最优线性码。定义了有限非链环[Fp+uFp+vFp]上[(1+u+v)]-循环码的相关概念,讨论了其生成多项式;引入了一种新的Gray映射[?],证明了该环上[(1+u+v)]-循环码在此Gray映射[?]下的[p]元象是一个长为[2pn]的2-准循环码,并由此构造出了两个最优二元准循环码。     

Recursive algorithms for linear LMSE estimators under uncertain observations

Hadidi and Schwartz [1] proposed a recursive linear estimator algorithm for linear discrete-time systems under uncertain observations when it exists. In this note two sets of formulas for the optimal filter and one-step predictor under uncertain observations, when they exist, are derived. These formulas differ, to some extent, from those proposed in [1].     

On optimal parameter selection for parabolic differential systems

In ordinary differential systems, it has been shown [4] recently that Boltyanskii's necessary conditions for optimal parameter selection [3, p. 263] can be easily derived from Gamkrelidze's general maximum principle [2]. In this note, necessary conditions on optimal parameter selection problem of parabolic differential systems are reported by using a recent result of Zolezzi [1]. The result is illustrated by an example.     

On the existence of optimal fixpoints

The concept of optimal fixpoint was introduced by Manna and Shamir [6, 7] in order to extract the maximum amount of “useful” information from a recursive definition. In this paper, we extend the concept of optimal fixpoint to arbitrary posets and investigate conditions which guarantee their existence. We prove that if a poset is chain-complete and has bounded joins, then every monotonic function has an optimal fixpoint. We also provide a sort of converse which generalizes a Theorem of A. Davis [2]. If a lower semilattice has bounded joins and every monotonic function has a fixpoint, then it is chain-complete.     

Remarks on two fixed-point theorems involving the sum and the product of two operators

In this article, we focus our attention on two fixed-point theorems of Krasnoselskii [1] and Dhage [2]. It is shown that some of the hypotheses of these fixed-point theorems are redundent. Our claim is also illustrated with an example.     

Addendum to “Code spectrum and the reliability function: Binary symmetric channel”

A much simpler proof of Theorem 1 from [1] is presented; we use the notation and enumeration of formulas of [1]. The text below replaces the subsection General Case in [1, Section 4, p. 271].     

Computing fourier integrals by means of near-optimal quadrature rules of the Lagrangian type

As shown inGustafson-Dahlquist [3] the computation of a large class of slowly convergent Fourier integrals can be cast into the problem of evaluating a certain Stieltjes integral over the interval [0, 1]. If the integrator can be shown to be non-decreasing, generalized Gauss rules can be used but this calls for the solution of a generally ill-conditioned non-linear system. In this paper we report experimental results, which illustrate the fact, that if the abscissae are chosen according to a simple strategy and then the weights are computed from a linear system, the resulting rules of Lagrangian type are almost optimal in a certain sense.     

The stability of long action chains in XCS

 XCS [1, 2] represents a new form of learning classifier system [3] that uses accuracy as a means of guiding fitness for selection within a Genetic Algorithm. The combination of accuracy-based selection and a dynamic niche-based deletion mechanism achieve a long sought-after goal–the reliable production, maintenance, and proliferation of the sub-population of optimally general accurate classifiers that map the problem domain [4]. Wilson [2] and Lanzi [5, 6] have demonstrated the applicability of XCS to the identification of the optimal action-chain leading to the optimum trade-off between reward distance and magnitude. However, Lanzi [6] also demonstrated that XCS has difficulty in finding an optimal solution to the long action-chain environment Woods-14 [7]. Whilst these findings have shed some light on the ability of XCS to form long action-chains, they have not provided a systematic and, above all, controlled investigation of the limits of XCS learning within multiple-step environments. In this investigation a set of confounding variables in such problems are identified. These are controlled using carefully constructed FSW environments [8, 9] of increasing length. Whilst investigations demonstrate that XCS is able to establish the optimal sub-population [O] [4] when generalisation is not used, it is shown that the introduction of generalisation introduces low bounds on the length of action-chains that can be identified and chosen between to find the optimal pathway. Where these bounds are reached a form of over-generalisation caused by the formation of dominant classifiers can occur. This form is further investigated and the Domination Hypothesis introduced to explain its formation and preservation.     

On the type of linear multivariable systems

Based on the prime factorization of transfer function matrices, alternative proofs of some basic properties of the type of linear multivariable systems [1] are given. Also, this leads to strengthening some results given in [1]. A connection is made between the system type and the Smith-McMillan form of the transfer function. An error in an example used in [1] is pointed out.     

Comments on "The design of linear regulators optimal for time multiplied performance indices"

A procedure for the design of linear regulators optimal with respect to a time-multiplied performance index was suggested in a recent paper by Man and Smith. A counterexample, presented subsequently [2], indicated that the procedure was in error. It is shown that the optimal feedback control law must contain time-varying gains as opposed to the time-invariant gains as suggested by Man and Smith. This fact was also overlooked in [2].     

Comments on "A nonlinear economic control problem with a linear feedback solution"

In a recent paper by Luenberger[1] it was shown that a special nonlinear economic control problem had a unique global solution expressible in linear feedback form and that the optimal return function was affine. It is shown here that these results may be extended to a much wider class of economic problems.     

On the solution of interval linear systems

In the literature efficient algorithms have been described for calculating guaranteed inclusions for the solution of a number of standard numerical problems [3,4,8,11,12,13]. The inclusions are given by means of a set containing the solution. In [12,13] this set is calculated using an affine iteration which is stopped when a nonempty and compact set is mapped into itself. For exactly given input data (point data) it has been shown that this iteration stops if and only if the iteration matrix is convergent (cf. [13]). In this paper we give a necessary and sufficient stopping criterion for the above mentioned iteration for interval input data and interval operations. Stopping is equivalent to the fact that the algorithm presented in [12] for solving interval linear systems computes an inclusion of the solution. An algorithm given by Neumaier is discussed and an algorithm is proposed combining the advantages of our algorithm and a modification of Neumaier's. The combined algorithm yields tight bounds for input intervals of small and large diameter. Using a paper by Jansson [6,7] we give a quite different geometrical interpretation of inclusion methods. It can be shown that our inclusion methods are optimal in a specified geometrical sense. For another class of sets, for standard simplices, we give some interesting examples.     

The regularization ofNB-scheme in differential games

The optimality principles in non-zero sum differential games and multicriterial control problems taken from the corresponding static (simultaneous) game theory are usually dynamic unstable (time inconsistent), thus their use becomes questionable if special regularization attempts are not made. This important aspect was first considered in [1], but also in the earlier paper of Strotz [7] this was shown for a special control problem with discount payoff. We have shown in [2] that many of the known procedures (excluding those based on scalarization of the payoff vector), of selecting of a special Pareto-optimal solution from the set of all Pareto-optimal solutions are dynamic unstable (time-inconsistent) and therefore the regularization procedures leading to the dynamic stable optimality principles are purposed [3]. It turns out that also in the Nash [4] bargaining process the regularization attempt can be made by constructng a special equation for the conflict point. The idea of the method is closely connected with the paper [5]. The special attention to the time-consistency problem is given also in the recent publications [6,8].This paper is financially supported by Russian Fundamental Research Foundation (project N 93-011-1714).     

An optimal linear operator for step edge detection

Step edge detection is an important subject in image processing and computer vision and many methods, including some optimal filters, have been proposed. In this paper, we propose an optimal linear operator of an infinite window size for step edge detection. This operator is at first derived from the well-known mono-step edge model by use of a signal/noise ratio adapted to edge detection. Because of the infinite window size of the operator, we propose then a statistic multiedge model and analyze the optimal operator by spectral analysis. It is shown that the Infinite Symmetric Exponential Filter (ISEF) is optimal for both mono- and multiedge detection. Recursive realization of ISEF and the derivatives is presented and generalized to multidimensional cases also. The performance of ISEF is analyzed and compared with that of Gaussian and Canny filters, and it is shown that ISEF has a better performance in precision of edge localization, insensibility to noise, and computational complexity. Edge detection based on the optimal filter ISEF is thus presented and the essential difference between ISEF and some other optimal edge detectors is shown. The experimental results for computer-generated and real images, which confirm our theoretical analysis, are reported.     

On the pre-AFL of [log n] space and related families of languages

New insight into the 1965 conjecture of Lewis et al., that some Context-Free Languages (CFLs) require more than [log n] space for recognition on off-line Turing machines, is derived from an examination of the pre-AFL properties of [log n] space. General results about related AFLs are used to reduce this conjecture to the questions of whether the [log n] class is an AFL and of whether [log n] recognition is a decidable question for certain related AFLs. One of these is the AFL generated by the [log n] class itself, which AFL is shown to properly contain all CFLs (Theorem 1) and, also, to be generated via length-preserving homomorphisms from the [log n] class, using the result that the latter is a pre-AFL (Theorem 2).An example of a related family, which happens to be contained in this newly studied AFL, is the principal AFL, Q, of quasi-real-time languages. Also, the open question of whether [log n] space is the same deterministically or nondeterministically is related to the above questions.     

The optimal projection equations for model reduction and the relationships among the methods of Wilson, Skelton, and Moore

First-order necessary conditions for quadratically optimal reduced-order modeling of linear time-invariant systems are derived in the form of a pair of modified Lyapunov equations coupled by an oblique projection which determines the optimal reduced-order model. This form of the necessary conditions considerably simplifies previous results of Wilson [1] and clearly demonstrates the quadratic extremality and nonoptimality of the balancing method of Moore [2]. The possible existence of multiple solutions of the optimal projection equations is demonstrated and a relaxation-type algorithm is proposed for computing these local extrema. A component-cost analysis of the model-error criterion similar to the approach of Skelton [3] is utilized at each iteration to direct the algorithm to the global minimum.     

On the Butcher group and general multi-value methods

This paper proves a theorem (“Theorem 6”) on the composition of, what we call, Butcher series. This Theorem is shown to be fundamental for the theory of Runge-Kutta methods: the formulas for the Taylor expansion of RK-methods and multiderivative RK-methods as well as formulas for the operation of the “Butcher group” (which describes the composition of RK-methods) are easy consequences. We do not attempt to realize the series as (generalized) Runge-Kutta methods, so we are not forced to restrict ourselves to the finite dimensional case. The theory extends to the multiderivative case as well, and the formulas remain valid for series which are not realizable as Runge-Kutta methods at all. Finally we extend the multi-value methods of J. Butcher [2] to the multiderivative case, which leads to a big class of integration methods for ordinary differential equations, including the methods of Nordsieck and Gear [3]. The defintions and notations of [4] are used throughout this paper, many of the results are proved here again.     

Decentralized control in packet switched satellite communication

The model introduced by Schoute [1] is analyzed using a minimum principle for decentralized stochastic control [2]. For certain values of the input message ratespand numberNof transmitting stations, simple open-loop strategies are shown to be person-by-person optimal. The true optimal control is shown to be open loop forN = 2.