LIPSCHITZ SEMISTABILITY OF GENERAL CONTROL SYSTEMS IN TERMS OF TWO MEASURES

1 IntroductionLipschitz semistability of dynamical systems in terms of two measures andLipschitz semistability of general control systems have already been,t.di.d[llt12].In this papers semistability considerations will be extended to a variety of nonlinear systems utilizing the same versatile tools, namely, Liapunov--like functions. The notions of (ho, h)-Lipschitz (weak) semistability of general controlsystems are introduced, and the necessary and sufficient conditions are given.2 Prelimina…     

Finite-time semistability,Filippov systems,and consensus protocols for nonlinear dynamical networks with switching topologies

This paper focuses on semistability and finite-time semistability for discontinuous dynamical systems. Semistability is the property whereby the solutions of a dynamical system converge to Lyapunov stable equilibrium points determined by the system initial conditions. In this paper, we extend the theory of semistability to discontinuous autonomous dynamical systems. In particular, Lyapunov-based tests for strong and weak semistability as well as finite-time semistability for autonomous differential inclusions are established. Using these results we then develop a framework for designing semistable and finite-time semistable protocols for dynamical networks with switching topologies. Specifically, we present distributed nonlinear static and dynamic output feedback controller architectures for multiagent network consensus and rendezvous with dynamically changing communication topologies.     

Remarks on Semistability of G-Bundles in Positive Characteristic

We analyzes a notion of strong semistability of principal G-bundles by including reduction to nonreduced parabolic subgroup schemes. It turns out that strong semistability is equivalent to the Frobenius semistability of Ramanan and Rananathan. We also give a bound for nonstrongly semsitability of a semistable GL(n)-bundle improving a previous result of Shepherd-Barron.     

Asymptotic error distribution for the Euler scheme with locally Lipschitz coefficients

In traditional works on numerical schemes for solving stochastic differential equations (SDEs), the globally Lipschitz assumption is often assumed to ensure different types of convergence. In practice, this is often too strong a condition. Brownian motion driven SDEs used in applications sometimes have coefficients which are only Lipschitz on compact sets, but the paths of the SDE solutions can be arbitrarily large. In this paper, we prove convergence in probability and a weak convergence result under a less restrictive assumption, that is, locally Lipschitz and with no finite time explosion. We prove if a numerical scheme converges in probability uniformly on any compact time set (UCP) with a certain rate under a global Lipschitz condition, then the UCP with the same rate holds when a globally Lipschitz condition is replaced with a locally Lipschitz plus no finite explosion condition. For the Euler scheme, weak convergence of the error process is also established. The main contribution of this paper is the proof of $\sqrt{n}$ weak convergence of the normalized error process and the limit process is also provided. We further study the boundedness of the second moments of the weak limit process and its running supremum under both global Lipschitz and locally Lipschitz conditions.     

Semistability of switched dynamical systems,Part I: Linear system theory

This paper develops semistability and uniform semistability analysis results for switched linear systems. Semistability is the property whereby the solutions of a dynamical system converge to Lyapunov stable equilibrium points determined by the system’s initial conditions. Since solutions to switched systems are a function of the system’s initial conditions as well as the switching signals, uniformity here refers to the convergence rate of the multiple solutions as the switching signal evolves over a given switching set. The main results of the paper involve sufficient conditions for semistability and uniform semistability using multiple Lyapunov functions and sufficient regularity assumptions on the class of switching signals considered.     

Semistability of switched dynamical systems,Part II: Non-linear system theory

In a companion paper [Q. Hui, W.M. Haddad, Semistability of switched dynamical systems. Part I: Linear system theory, Non-linear Anal. Hybrid Syst. 3 (3) (2009) 343–353] semistability and uniform semistability results for switched linear systems were developed. In this paper we develop semistability analysis results for non-linear switched systems. Semistability is the property whereby the solutions of a dynamical system converge to Lyapunov stable equilibrium points determined by the system initial conditions. The main results of the paper involve sufficient conditions for semistability using multiple Lyapunov functions and integral-type inequalities.     

Weakly upper Lipschitz multifunctions and applications in parametric optimization

The main purpose of this paper is to report on our studies of the weak upper Lipschitz and weak -upper Lipschitz continuities of multifunctions. Comparisons with other related Lipschitz-type continuities and calmness are given. When the concept of the weak upper Lipschitz continuities is applied to the special cases of constraint multifunctions, such as ones defined by a systems of equalities and inequalities or by a generalized equation we obtain the equivalent conditions with linear functional error bounds. Some results on the perturbation and penalty issues in parametric optimization problems are obtained under weak upper Lipschitz continuity assumptions on the constraint multifunctions. We also discuss the weak -upper Lipschitz continuity of a inverse subdifferential.Mathematics Subject Classification (2000): 49J52, 49J53, 90C25Acknowledgement The author thanks the associate editor and the referees for their helpful suggestions and comments.     

Approximate Hermitian–Yang–Mills structures and semistability for Higgs bundles. I: generalities and the one-dimensional case

We review the notions of (weak) Hermitian–Yang–Mills structure and approximate Hermitian–Yang–Mills structure for Higgs bundles. Then, we construct the Donaldson functional for Higgs bundles over compact K?hler manifolds and we present some basic properties of it. In particular, we show that its gradient flow can be written in terms of the mean curvature of the Hitchin–Simpson connection. We also study some properties of the solutions of the evolution equation associated with that functional. Next, we study the problem of the existence of approximate Hermitian–Yang–Mills structures and its relation with the algebro-geometric notion of semistability and we show that for a compact Riemann surface, the notion of approximate Hermitian–Yang–Mills structure is in fact the differential- geometric counterpart of the notion of semistability. Finally, we review the notion of admissible Hermitian structure on a torsion-free Higgs sheaf and define the Donaldson functional for such an object.     

LIPSCHITZ STABILITY OF GENERAL CONTROL SYSTEMS

ＬＩＰＳＣＨＩＴＺＳＴＡＢＩＬＩＴＹＯＦＧＥＮＥＲＡＬＣＯＮＴＲＯＬＳＹＳＴＥＭＳＪｉｎＣａｎ（金淦）（ＺｈｏｎｇｓｈａｎＵｎｉｖｅｒｓｉｔｙ，中山大学，邮编：５１０２７５）Ａｂｓｔｒａｃｔ：Ｎｅｃｅｓｓａｒｙａｎｄｓｕｆｆｉｃｉｅｎｔｃｏｎｄｉｔｉ...     

Lipschitz compact operators

We introduce the notion of Lipschitz compact (weakly compact, finite-rank, approximable) operators from a pointed metric space X into a Banach space E. We prove that every strongly Lipschitz p-nuclear operator is Lipschitz compact and every strongly Lipschitz p-integral operator is Lipschitz weakly compact. A theory of Lipschitz compact (weakly compact, finite-rank) operators which closely parallels the theory for linear operators is developed. In terms of the Lipschitz transpose map of a Lipschitz operator, we state Lipschitz versions of Schauder type theorems on the (weak) compactness of the adjoint of a (weakly) compact linear operator.     

Existence and uniqueness of the solutions of stochastic differential equations

The standard existence and uniqueness theorem for stochastic differential equations requires Lipschitz condition of the coefficients. In this paper, we extend these results to the case in which the coefficients are not required to be Lipschitz continuous, instead they only satisfy a ‘weak’ type of Lipschitz condition.     

The computable Lipschitz degrees of computably enumerable sets are not dense

The computable Lipschitz reducibility was introduced by Downey, Hirschfeldt and LaForte under the name of strong weak truth-table reducibility (Downey et al. (2004) [6]). This reducibility measures both the relative randomness and the relative computational power of real numbers. This paper proves that the computable Lipschitz degrees of computably enumerable sets are not dense. An immediate corollary is that the Solovay degrees of strongly c.e. reals are not dense. There are similarities to Barmpalias and Lewis’ proof that the identity bounded Turing degrees of c.e. sets are not dense (George Barmpalias, Andrew E.M. Lewis (2006) [2]), however the problem for the computable Lipschitz degrees is more complex.     

一类Lipschitz B-(p,γ)-不变凸函数与非光滑规划

设本文给出了一类新的Lipschitz B-(p,γ)-不变凸函数,它是B-不变凸函数和(p,γ)-不变凸函数的推广.在这类Lipschitz B-(p,γ)一不变凸性下,建立了非光滑规划的必要和充分最优性条件,讨论了Mond-Weir型对偶和Wolfe型对偶,证明了弱对偶、强对偶和逆对偶定理.所得结果推广了涉及凸函数、B-不变凸函数和(p,γ)-不变凸函数的规划问题的相应结果.     

平面上全局半稳定的广义Liénard系统

该文研究了广义Lienard系统的全局半稳定,获得了系统（E）为全局半稳定系统的充要条件．     

动力系统在两种尺度意义下的Lipschitz半稳定性

本文将半稳定性（即在某种限制条件下的稳定性）和Ｌｉｐｓｃｈｉｔｚ稳定性,两尺度稳定性概念结合,提出动力系统的（ｈ０,ｈ）－半稳定和（ｈ０,ｈ）－Ｌｉｐｓｃｈｉｔｚ（局部）半稳定性概念,并且Ｌｉａｐｕｎｏ类似函数给出相应的充要条件。     

LOCAL CONVERGENCE OF INEXACT NEWTON-LIKE METHOD UNDER WEAK LIPSCHITZ CONDITIONS

The paper develops the local convergence of Inexact Newton-Like Method(INLM)for approximating solutions of nonlinear equations in Banach space setting. We employ weak Lipschitz and center-weak Lipschitz conditions to perform the error analysis. The obtained results compare favorably with earlier ones such as [7, 13, 14, 18, 19]. A numerical example is also provided.     

Carleman estimates for parabolic equations with interior degeneracy and Neumann boundary conditions

Density of Lipschitz functions in Newtonian spaces based on quasi-Banach function lattices is discussed. Newtonian spaces are first-order Sobolevtype spaces on abstract metric measure spaces defined via (weak) upper gradients. Our main focus lies on metric spaces with a doubling measure that support a Poincaré inequality. Absolute continuity of the function lattice quasi-norm is shown to be crucial for approximability by (locally) Lipschitz functions. The proof of the density result uses, among other facts, the fact that a suitable maximal operator is locally weakly bounded. In particular, various sufficient conditions for such boundedness on quasi-Banach function lattices (and rearrangement-invariant spaces, in particular) are established and applied.     

Harder–Narasimhan theory for linear codes (with an appendix on Riemann–Roch theory)

In this text we develop some aspects of Harder–Narasimhan theory, slopes, semistability and canonical filtration, in the setting of combinatorial lattices. Of noticeable importance is the Harder–Narasimhan structure associated to a Galois connection between two lattices. It applies, in particular, to matroids.We then specialize this to linear codes. This could be done from at least three different approaches: using the sphere-packing analogy, or the geometric view, or the Galois connection construction just introduced. A remarkable fact is that these all lead to the same notion of semistability and canonical filtration. Relations to previous propositions toward a classification of codes, and to Wei's generalized Hamming weight hierarchy, are also discussed.Last, we study the two important questions of the preservation of semistability (or more generally the behavior of slopes) under duality, and under tensor product. The former essentially follows from Wei's duality theorem for higher weights—and its matroid version—which we revisit in an appendix, developing analogues of the Riemann–Roch, Serre duality, Clifford, and gap and gonality sequence theorems. Likewise the latter is closely related to the bound on higher weights of a tensor product, conjectured by Wei and Yang, and proved by Schaathun in the geometric language, which we reformulate directly in terms of codes. From this material we then derive semistability of tensor product.     

动力系统中的Lipschitz稳定性

本文讨论动力系统的Ｌｉｐｓｃｈｉｔｚ稳定性与吸引性之间的关系．给出动力系统中弱吸引性、吸引性和强吸引性这三个概念相互等价的条件，在一定的条件下证明了（弱、强）吸引子与全局（弱、强）吸引子是一致的．本文还讨论了度量Ｌｙａｐｕｎｏｖ稳定性和拓扑Ｌｙａｐｕｎｏｖ稳定性及它们与Ｌｉｐｓｃｈｉｔｚ稳定性之间的关系．     

Necessary and Sufficient Conditions for Weak Quasi Efficiency in Nonsmooth Multiobjective Problems

In this article, a multiobjective problem with a feasible set defined by inequality, equality and set constraints is considered, where the objective and constraint functions are locally Lipschitz. Several constraint qualifications are given and the relations between them are analyzed. We establish Kuhn-Tucker and strong Kuhn-Tucker necessary optimality conditions for (weak) quasi e?ciency in terms of the Clarke subdifferential. By using two new classes of generalized convex functions, su?cient conditions for local (weak) quasi e?cient are also provided. Furthermore, we study the Mond-Weir type dual problem and establish weak, strong and converse duality results.