Some Aspects of Synchronization of DFA

A word w is called synchronizing （recurrent, reset, directable） word of deterministic finite automata （DFA） if w brings all states of the automaton to a unique state. According to the famous conjecture of Cerny from 1964, every n-state synchronizing automaton possesses a synchronizing word of length at most （n - 1）2. The problem is still open. It will be proved that the Cerny conjecture holds good for synchronizing DFA with transition monoid having no involutions and for every n-state （n 〉 2） synchronizing DFA with transition monoid having only trivial subgroups the minimal length of synchronizing word is not greater than （n - 1）2/2. The last important class of DFA involved and studied by Schutzenberger is called aperiodic; its automata accept precisely star-free languages. Some properties of an arbitrary synchronizing DFA were established.     

一种半群序列的一致预言定理

给出了一点代数几何码的伴随式半群序列的概念,定义了这种序列的一种递推关系,建立了这种递推关系的一致预言定理。     

Approximations in <Emphasis Type="Italic">n</Emphasis>-ary algebraic systems

Algebraic systems have many applications in the theory of sequential machines, formal languages, computer arithmetics, design of fast adders and error-correcting codes. The theory of rough sets has emerged as another major mathematical approach for managing uncertainty that arises from inexact, noisy, or incomplete information. This paper is devoted to the discussion of the relationship between algebraic systems, rough sets and fuzzy rough set models. We shall restrict ourselves to algebraic systems with one n-ary operation and we investigate some properties of approximations of n-ary semigroups. We introduce the notion of rough system in an n-ary semigroup. Fuzzy sets, a generalization of classical sets, are considered as mathematical tools to model the vagueness present in rough systems.     

基于谱分解的中立型延时系统控制器设计

针对线性中立型延时系统, 提出一种基于算子半群谱分解理论的控制器设计方法. 在中立型项的范数小于1的情况下, 利用谱理论并结合投影算子, 将系统无穷维解空间分解为有限维不稳定广义特征子空间和无限维稳定子空间的直和. 通过对有限维子系统的研究, 得到系统状态反馈控制律, 并通过投影算子证明了状态反馈控制闭环系统的渐近稳定性. 最后数值例子验证了所提方法的有效性.     

Characterizations of regular ordered semigroups in terms of (α, β)-fuzzy generalized bi-ideals

In this paper, the concept of an (α, β)-fuzzy generalized bi-ideal in an ordered semigroup is introduced, which is a generalization of the concept of a fuzzy generalized bi-ideal in an ordered semigroup. Using this concept, some characterization theorems are provided. The upper/lower parts of an (∈, ∈ ∨ q)-fuzzy generalized bi-ideal are introduced and some characterizations of regular ordered semigroups are given. Also, we consider the concept of implication-based fuzzy generalized bi-ideals in an ordered semigroup. In particular, the implication operators in Lukasiewicz system of continuous-valued logic are discussed.     

半群中的(λ,μ)-模糊双理想与(λ,μ)-模糊拟理想

在一个半群中引入了(λμ)-模糊双理想和(λμ)-模糊拟理想的概念,研究了这些理想的基本性质,并给出了它们的一些等价条件。最后,利用(λμ)-模糊双理想和(λμ)-模糊拟理想,刻画了正则半群的特征。     

Admissible Observation Operators for the Right-Shift Semigroup

We give a characterization of infinite-time admissible observation operators for the right-shift semigroup on L ²[0,∞). Our main result is that if A is the generator of this semigroup and C is the observation operator, mapping D(A) into the complex numbers, then C is infinite-time admissible if and only if for all s in the open right half-plane. We derive this using Fefferman's theorem on bounded mean oscillation and Hankel operators. This result solves a special case of a more general conjecture which says that the same equivalence is true for any strongly continuous semigroup acting on a Hilbert space. For normal semigroups the conjecture is known to be true and then it is equivalent to the Carleson measure theorem. We derive some related results and partial results concerning the case when the signals are not scalar but with values in a Hilbert space. Date received: January 15, 1999. Date revised: September 24, 1999.     

A partial history of the early development of continuous-time nonlinear stochastic systems theory

This article is a survey of the early development of selected areas in nonlinear continuous-time stochastic control. Key developments in optimal control and the dynamic programming principle, existence of optimal controls under complete and partial observations, nonlinear filtering, stochastic stability, the stochastic maximum principle and ergodic control are discussed. Issues concerning wide bandwidth noise for stability, modeling, filtering and ergodic control are dealt with. The focus is on the earlier work, but many important topics are omitted for lack of space.     

Spectral properties of infinite-dimensional closed-loop systems

We consider feedback systems obtained from infinite-dimensional well-posed linear systems by output feedback. Thus, our framework allows for unbounded control and observation operators. Our aim is to investigate the relationship between the open-loop system, the feedback operator K and the spectrum (in particular, the eigenvalues and eigenvectors) of the closed-loop generator A^K. We give a useful characterization of that part of the spectrum σ(A^K) which lies in the resolvent set of A, the open-loop generator, via the “characteristic equation” involving the open-loop transfer function. We show that certain points of σ(A) cannot be eigenvalues of A^K if the input and output are scalar (so that K is a number) and K≠0. We devote special attention to the case when the output feedback operator K is compact. It is relatively easy to prove that in this case, σ_e(A), the essential spectrum of A, is invariant, that is, it is equal to σ_e(A^K). A related but much harder problem is to determine the largest subset of σ(A) which remains invariant under compact output feedback. This set, which we call the immovable spectrum of A, strictly contains σ_e(A). We give an explicit characterization of the immovable spectrum and we investigate the consequences of our results for a certain class of well-posed systems obtained by interconnecting an infinite chain of identical systems.     

The Asymptotic Behavior for the Streaming Operator in Slab Geometry

In this paper, we complete Generation theorem for the streaming operator in slab geometry, where we have proved that the one-dimensional streaming operator in slab domain, with general boundary conditions described by a linear operator K, generates a strongly continuous semigroup. Here, under the positivity and irreducibility of the boundary operator K, we prove the positivity and the irreducibility of the generated semigroup. We also study the spectral properties of the streaming operator and characterize its spectral bound. Under the compactness of the boundary operator K, we determine the essential type of the generated semigroup and find its behavior asymptotic in the operator norm topology.