Uniform Stability for Time-Varying Infinite-Dimensional Discrete Linear Systems

Stability for time-varying discrete linear systems in a Banachspace is investigated. On the one hand is established a fairlycomplete collection of necessary and sufficient conditions foruniform asymptotic equistability for input-free systems. Thisincludes uniform and strong power equistability, and uniformand strong l_p-equistability, among other technical conditionswhich also play an essential role in stability theory. On theother hand, it is shown that uniform asymptotic equistabilityfor input-free systems is equivalent to each of the followingconcepts of uniform stability for forced systems: l_p-input l_p-state,e_o-input e_o-state, bounded-input bounded-state, l_p-input bounded-state(with p>1), e_o-input bounded-state, and convergent-inputbounded-state; these are also equivalent to their nonuniformcounterparts. For time-varying convergent systems, the aboveis also equivalent to convergent-input convergent-state stability.The proofs presented here are all ‘lementary’ inthe sense that they are based essentially only on the Banach–Steinhaustheorem.     

Nonsmooth Mapping of Linear Control Systems

This paper deals with linear control systems of a special form. The main goal of the paper is to find the exact analytic solution of the time-optimal control problem for an arbitrary linear control system with constant coefficients using the analytic solution of this problem for the canonical system. For this aim, we construct a certain nonsmooth mapping between the 0-controllability sets of the given systems. In other words, by means of this mapping, we investigate the equivalence of the systems with the same qualitative behavior in a neighborhood of the stationary point.     

Torsion-free path geometries and integrable second order ODE systems

A search for invariants of second order ODE systems under the class of point transformations, which mix the parameter and the dependent variables, uncovers a torsion tensor generalizing part of the curvature tensor of an affine connection. We study the geometry of ODE systems for which this torsion vanishes. These are the ODE systems for which deformations of solutions fixing a point constitute a field of Segré varieties in the tangent bundle of the locally defined space of solutions. Conversely, a field of Segré varieties for which certain differential invariants vanish induces a torsion-free ODE system on the space of solutions to a natural PDE system. The geometry on the solution space is used to produce first integrals for torsion-free ODE systems, given as algebraic invariants of a curvature tensor involving up to fourth derivatives of the equations. In the generic case, there are enough first integrals to solve the equations explicitly in spite of the absence of symmetry. In the case of torsion-free ODE pairs, the field of Segré varieties is equivalent to a half-flat split signature conformal structure, and we characterize in terms of curvature those systems having an abundance of totally geodesic surfaces.     

The Lie Theory of Two-Variable Hypergeometric Functions

We develop a Lie-algebraic method that associates with each of the 34 distinct second-order hypergeometric functions in two variables a canonical system of partial differential equations. The special functions arise by partial separation of variables in these simple systems. Some consequences are a demonstration that all such functions appear as solutions of the 4-variable wave equation and a classification of the possible imbeddings. In each case the functions are characterized by first- and second-order operators in the enveloping algebra of the conformal symmetry algebra for the wave equation. In some cases the 3-variable wave and heat equations and the 2-variable Helmholtz equation also arise. This intimate relationship between Horn functions and some fundamental equations of mathematical physics shows that these functions are more interesting than was previously recognized and permits use of the powerful tools of Lie theory and separation of variables to obtain properties of the functions.     

Structure of coordinate groups for algebraic sets in partially commutative nilpotent groups

The results obtained deal in algebraic geometry over partially commutative class two nilpotent ℚ-groups, where ℚ is a field of rationals. It is proved that two arbitrary non-Abelian partially commutative class two nilpotent ℚ-groups are geometrically equivalent. A necessary and sufficient condition of being universally geometrically equivalent is specified for two partially commutative class two nilpotent ℚ-groups. Algebraic sets for systems of equations in one variable, as well as for some special systems in several variables, are described. Dedicated to V. N. Remeslennikov on the occasion of his 70th birthday Translated from Algebra i Logika, Vol. 48, No. 2, pp. 378–399, May–June, 2009.     

An Iterative Analytical Theory in the Mechanics of Layered Composite Systems

An iterative analytical theory in the mechanics of layered composite systems is developed. The prehistory of the nonclassical theory of layered systems is presented. The division of this theory into two principal directions - discrete-structural and continuous-structural - is mentioned. The basic iterative Ambartsumyan theory, which belongs to the second direction, is described. The formation of the generalized iteration theory of first approximation is shown. In this theory, the disagreement between the kinematic and static models is removed, i.e., a generalization of these models is realized. The theory of second approximation is described. An iterative principle is presented for the formation of a higher-approximation nonclassical theory. Based on this principle, theories of anisotropic composite shallow shells, plates, and beams are formulated. Comparative calculation results for different layered composite systems are presented.     

Integrability of the Equations for Nonsingular Pairs of Compatible Flat Metrics

We solve the problem of describing all nonsingular pairs of compatible flat metrics (or, in other words, nonsingular flat pencils of metrics) in the general N-component case. This problem is equivalent to the problem of describing all compatible Dubrovin–Novikov brackets (compatible nondegenerate local Poisson brackets of hydrodynamic type) playing an important role in the theory of integrable systems of hydrodynamic type and also in modern differential geometry and field theory. We prove that all nonsingular pairs of compatible flat metrics are described by a system of nonlinear differential equations that is a special nonlinear differential reduction of the classical Lamé equations, and we present a scheme for integrating this system by the method of the inverse scattering problem. The integration procedure is based on using the Zakharov method for integrating the Lamé equations (a version of the inverse scattering method).     

Fractal balls

To the best of our knowledge, the analysis of densely folded media has not deserved special attention. The stress and strain analysis of this type of structures involves considerable difficulties concerning very strong non-linear effects. This paper presents a theory that could be classified as a geometric theory of folded media, in the sense that it ultimately leads to a kind of geometric constitutive law, or, in other words, a law that establishes the relationship between the geometry of the folded media and other variables such as the confinement capacity and the plastic strain energy. The discussion presented here is restricted to the particular case of compact balls produced by crushing together very thin plates or sheets. It is shown that both the geometry of the folded sheet and the plastic work density can be used as self-similarity tests. These criteria are equivalent for the case of thin plates or sheets made of the same material and with the same thickness. For the general case, the geometry of the folded sheet is not valid anymore as similarity criterion but there are strong arguments in favor of the plastic work density as a general criterion. If self-similarity is obtained for a ball set resulting from crumpling thin plates or sheets, it is possible to define two variables, the packing capacity and the slenderness ratio, that are related according to a power law. That is, the balls have a fractal representation. The power law scaling is derived from the mass conservation principle. The theory is claimed to be valid provided that certain assumptions referring to the geometry and material properties are satisfied. The results have shown that the theory is coherent and worthwhile of experimental validation. Some applications are suggested. A possible challenging investigation is related to the optimal geometry of biological membranes.     

A New Geometric Approach to Lie Systems and Physical Applications

The characterization of systems of differential equations admitting a superposition function allowing us to write the general solution in terms of any fundamental set of particular solutions is discussed. These systems are shown to be related with equations on a Lie group and with some connections in fiber bundles. We develop two methods for dealing with such systems: the generalized Wei–Norman method and the reduction method, which are very useful when particular solutions of the original problem are known. The theory is illustrated with some applications in both classical and quantum mechanics.     

About the Sojourn Time Process in Multiphase Queueing Systems

Multiphase queueing systems (MQS) (tandem queues, queues in series) are of special interest both in theory and in practical applications (packet switch structures, cellular mobile networks, message switching systems, retransmission of video images, asembly lines, etc.). In this paper, we deal with approximations of MQS and present a heavy traffic limit theorems for the sojourn time of a customer in MQS. Functional limit theorems are proved for the customer sojourn time – an important probability characteristic of the queueing system under conditions of heavy traffic.      

On Convex Generalized Systems

In this paper, the set-convexity and mapping-convexity properties of the extended images of generalized systems are considered. By using these image properties and tools of topological linear spaces, separation schemes ensuring the impossibility of generalized systems are developed. Then, special problem classes are investigated.     

半拓扑系统

本文旨在建立Ｓ系统理论，它具有相当的广泛性，以及应用背景性，同时又以拓扑空间、模糊拓扑空间、拓扑分子格以及拓扑系统为特例．     

Qualitative analysis of systems with an ideal non-conservative constraint

The Hamiltonian form developed in /1/ for the equations of motion of systems with ideal non-conservative constraints enables familiar methods of classical and celestial mechanics to be used to analyse the dynamics of such systems. When this is done certain difficulties arise, due to the fact that the Hamiltonian is not analytic. In this paper one of the possible algorithms applying KAM theory /2/ and Poincaré's theory of periodic motions /3/ to the analysis of systems in which the Hamiltonian is non-analytic in one of the phase variables is described. As an example, some results of /4/ concerning the dynamics of a rigid body colliding with a fixed, absolutely smooth, horizontal plane are refined.     

Hamilton-Jacobi methods for Vakonomic Mechanics

We extend the theory of Aubry-Mather measures to Hamiltonian systems that arise in vakonomic mechanics and sub-Riemannian geometry. We use these measures to study the asymptotic behavior of (vakonomic) action-minimizing curves, and prove a bootstrapping result to study the partial regularity of solutions of convex, but not strictly convex, Hamilton-Jacobi equations.      

Reduction,Linearization, and Stability of Relative Equilibria for Mechanical Systems on Riemannian Manifolds

Consider a Riemannian manifold equipped with an infinitesimal isometry. For this setup, a unified treatment is provided, solely in the language of Riemannian geometry, of techniques in reduction, linearization, and stability of relative equilibria. In particular, for mechanical control systems, an explicit characterization is given for the manner in which reduction by an infinitesimal isometry, and linearization along a controlled trajectory “commute.” As part of the development, relationships are derived between the Jacobi equation of geodesic variation and concepts from reduction theory, such as the curvature of the mechanical connection and the effective potential. As an application of our techniques, fiber and base stability of relative equilibria are studied. The paper also serves as a tutorial of Riemannian geometric methods applicable in the intersection of mechanics and control theory. F. Bullo’s research supported in part by grant CMS 0442041 from the USA National Science Foundation. A.D. Lewis’ research supported in part by grant from the Natural Sciences and Engineering Research Council of Canada.     

辛映射与Hamilton控制系统

给定等价辛流形 ,即辛同态或形变等价的辛流形 ,研究了建立在这些辛流形上的Hamilton控制系统之间的一些性质的联系 ,诸如 (局部 )能观测性 ,强可接近性 ,(拟 )极小性等 .而且 ,利用Cort啨ｓ介绍的 (弱 )外等价系统的概念 ,给出使得两个Hamilton控制系统是辛同态的一个充分条件     

多步Runge-Kutta方法关于一类多延迟系统的GPk-稳定性

本文多步Ｒｕｎｇｅ－Ｋｕｔｔａ方法关于延迟微分方程系统的渐近稳定性,在本文中我们证明了在适当条件下常微多步Ｒｕｎｇｅ－Ｋｕｔｔａ方法的Ａ－稳定性等价于相应求解多延迟微分方程系统的ＧＰｋ－稳定性。     

Left-symmetric algebras, or pre-Lie algebras in geometry and physics

In this survey article we discuss the origin, theory and applications of left-symmetric algebras (LSAs in short) in geometry in physics. Recently Connes, Kreimer and Kontsevich have introduced LSAs in mathematical physics (QFT and renormalization theory), where the name pre-Lie algebras is used quite often. Already Cayley wrote about such algebras more than hundred years ago. Indeed, LSAs arise in many different areas of mathematics and physics. We attempt to give a survey of the fields where LSAs play an important role. Furthermore we study the algebraic theory of LSAs such as structure theory, radical theory, cohomology theory and the classification of simple LSAs. We also discuss applications to faithful Lie algebra representations.     

Structural Properties of Homomorphism Dilation Systems

Inspired by some recent development on the theory about projection valued dilations for operator valued measures or more generally bounded homomorphism dilations for bounded linear maps on Banach algebras, the authors explore a pure algebraic version of the dilation theory for linear systems acting on unital algebras and vector spaces. By introducing two natural dilation structures, namely the canonical and the universal dilation systems, they prove that every linearly minimal dilation is equivalent to a reduced homomorphism dilation of the universal dilation, and all the linearly minimal homomorphism dilations can be classified by the associated reduced subspaces contained in the kernel of synthesis operator for the universal dilation.     

Weak Factorization Systems and Topological Functors

Weak factorization systems, important in homotopy theory, are related to injective objects in comma-categories. Our main result is that full functors and topological functors form a weak factorization system in the category of small categories, and that this is not cofibrantly generated. We also present a weak factorization system on the category of posets which is not cofibrantly generated. No such weak factorization systems were known until recently. This answers an open problem posed by M. Hovey.