Controllability and extremality in nonconvex differential inclusions

LetF:[0, T]×R ⁿ 2 ^{R n} be a set-valued map with compact values; let :R ⁿ R ^m be a locally Lipschitzian map,z(t) a given trajectory, andR the reachable set atT of the differential inclusion . We prove sufficient conditions for (z(T))intR and establish necessary conditions in maximum principle form for (z(T))(R). As a consequence of these results, we show that every boundary trajectory is simultaneously a Pontryagin extremal, Lagrangian extremal, and relaxed Lagrangian extremal.The author is grateful to an anonymous referee for his valuable remarks and comments which have helped to improve the paper.The paper was written while the author was visiting the laboratory of Prof. S. Suzuki, Department of Mechanical Engineering, Sophia University, Tokyo, Japan.     

Iterative Schemes for Solving Mixed Variational-Like Inequalities

In the present paper, we introduce the concept of -cocoercivity of a map and develop some iterative schemes for finding the approximate solutions of mixed variational-like inequalities. We use the concept of -cocoercivity to prove the convergence of the approximate solutions to the exact solution of mixed variational-like inequalities.     

Variational-Like Inequalities with Generalized Monotone Mappings in Banach Spaces

In this paper, we introduce two classes of variational-like inequalities with generalized monotone mappings in Banach spaces. Using the KKM technique, we obtain the existence of solutions for variational-like inequalities with relaxed – monotone mappings in reflexive Banach spaces. We present also the solvability of variational-like inequalities with relaxed – semimonotone mappings in arbitrary Banach spaces by means of the Kakutani-Fan-Glicksberg fixed-point theorem. The results presented in this paper extend and improve the corresponding results of Refs. 1–6.     

(φ,ξ,η,λ)‐Structures of Parabolic Type

We consider (,,,)structures of parabolic type on hypersurfaces of dual spaces and study the rank of the affinor . We consider almost contact metric structures of parabolic type of the first kind on hypersurfaces of 4dimensional dual metric space. We study the properties of these structures and give examples of normal, integrable, and Sasakian parabolic structures.     

Multiplicative Products of Dedekind η-Functions and Group Representations

In this paper, we findall metacyclic groups (a,b:a^m=e,b^s=e,b^-1ab=a^r), where m=10,14,15,20,21,22, such that the cusp forms associated with all elements of these groups by an exact representation are multiplicative-products. We also consider the correspondence between multiplicative -products and elements of finite order in SL(5,C) by the adjoint representation.     

The η-Invariant of Twisted Dirac Operators of S3/Γ

The aim of this paper is to compute the and ~-invariants for the Dirac operator of the quotient of the sphere S ³ by a finite subgroup, twisted by a representation of its fundamental group.     

Explicit Formulas for Green’s Functions on the Annulus and on the Möbius Strip

Given a fixed point free antianalytic involution k of a domain G in thecomplex plane, bounded by a finite number of analytic curves, k-invariant Greensfunctions are defined on G. The Lindelöfs principle is extended to k-invariantGreens functions. When G is the annulus, k-invariant Greens functions areobtained in the explicit form. Since the factorization of the annulus by the group kgenerated by k produces a Möbius strip, the respective result helped us to obtain explicitforms for Greens functions on the Möbius strip.     

Start‐up class models in multiple‐class queues with N‐policy

In this paper, we consider multipleclass queueing systems with Npolicy in which the idle server starts service as soon as the number of customers in the startup class reaches threshold N. We consider the cases of FCFS and nonpreemptive priority. We obtain the Laplace–Stieltjes transform of the waiting times of each class of customers. We also show some results for the general behavior of such systems.     

Optimal ℋ∞-state feedback control for continuous-time linear systemsfeedback control for continuous-time linear systems

This paper proposes a convex programming method to achieve optimal -state feedback control for continuous-time linear systems. State space conditions, formulated in an appropriate parameter space, define a convex set containing all the stabilizing control gains that guarantee an upper bound on the -norm of the closed-loop transfer function. An optimization problem is then proposed, in order to minimize this upper bound over the previous convex set, furnishing the optimal -control gain as its optimal solution. A limiting bound for the optimum -norm can easily be calculated, and the proposed method will achieve minimum attenuation whenever a feasible state feedback controller exists. Generalizations to decentralized and output feedback control are also investigated. Numerical examples illustrate the theory.This research has been supported in part by grants from Fundação de Amparo à Pesquisa do Estado de São Paulo—FAPESP and Conselho Nacional de Desenvolvimento Científico e Tecnológico, CNPq—Brazil. The authors are grateful to the anonymous referees for their useful comments on this paper.     

Generalized γ-Valid Cut Procedure for Concave Minimization

The concept of a -valid cutting plane has been used in many types of algorithms for solving concave minimization problems. Unfortunately, the procedures proposed to date for constructing these cuts are valid only under certain assumptions that often may not hold in practice. Chief among these is the requirement that the feasible region of the concave minimization problem in question have full dimension, and that the objective function of this problem be concave rather than quasiconcave. In this article, we propose, validate, and show how to implement a more general -valid cutting plane procedure which eliminates these restrictions.     

Spatial shape‐preserving interpolation using ν‐splines

We present a global iterative algorithm for constructing spatial G ²continuous interpolating splines, which preserve the shape of the polygonal line that interpolates the given points. Furthermore, the algorithm can handle data exhibiting two kinds of degeneracy, namely, coplanar quadruples and collinear triplets of points. The convergence of the algorithm stems from the asymptotic properties of the curvature, torsion and Frénet frame of splines for large values of the tension parameters, which are thoroughly investigated and presented. The performance of our approach is tested on two data sets, one of synthetic nature and the other of industrial interest.     

Asymptotic Behaviour of λ-Convex Sets in the Hyperbolic Plane

It is known that the limit Area/Length for a sequence of convex sets expanding over the whole hyperbolic plane is less than or equal 1, and exactly 1 when the sets considered are convex with respect to horocycles. We consider geodesics and horocycles as particular cases of curves of constant geodesic curvature with 0 1 and we study the above limit Area/Length as a function of the parameter .     

On θ-Principal Extension and H-Closedness

A type of extensions called the -extention of topological spaces and their -equivalence and -trace systems are introduced, which ultimately characterize H-closed -extensions of a Hausdorff topological space. Also, the notion of -principal extensions is defined. A typical -principal extension consisting of certain grills on a Hausdorff space is constructed, and finally, some characterizations of H-closedness of a Hausdorff space are obtained.AMS Subject Classification (2000) 54D30 54D99     

On Some Isomorphism on the Category of b-Spaces

Given a nuclear b-space N, we show that if is a finite or -finite measure space and 1p, then the functors L _loc ^p (,N.) and NL ^p (,.) are isomorphic on the category of b-spaces of L. Waelbroeck.     

On Connectedness via Closure Operators

This paper describes a convenient modification of the approach presented in the paper Closure operators and connectedness by G. Castellini and D. Hajek, which is shown to give a suitable generalization of left- and right-constant subcategories, both at the object and the morphism levels. We show in particular that the framework we introduce here allows the simultaneous study of the classes of connected topological spaces, of concordant continuous maps and of monotone continuous maps.     

On the Movement of Transitive Permutation p-Groups

Let G be a transitive permutation group on a set and m a positive integer. If | – | m for every subset of and all g G, then || 2mp/(p – 1) where p is the least odd prime dividing |G|. It was shown by Mann and Praeger [13] that, for p = 3, the 3-groups G which attain this bound have exponent p. In this paper we will show a generalization of this result for any odd primes.AMS Subject Classification (2000), 20BXX     

A Stationary Rho-Mixing Markov Chain Which Is Not “Interlaced” Rho-Mixing

A strictly stationary, countable-state Markov chain is constructed which is -mixing (with arbitrarily fast mixing rate) but fails to be *-mixing (interlaced-mixing).     

Piecewise Constant Roughly Convex Functions

This paper investigates some kinds of roughly convex functions, namely functions having one of the following properties: -convexity (in the sense of Klötzler and Hartwig), -convexity and midpoint -convexity (in the sense of Hu, Klee, and Larman), -convexity and midpoint -convexity (in the sense of Phu). Some weaker but equivalent conditions for these kinds of roughly convex functions are stated. In particular, piecewise constant functions satisfying f(x) = f([x]) are considered, where [x] denotes the integer part of the real number x. These functions appear in numerical calculation, when an original function g is replaced by f(x):=g([x]) because of discretization. In the present paper, we answer the question of when and in what sense such a function f is roughly convex.