Triangulations and the stability theorem for foliations

Introduction. 1.1 Linear Time-Invariant Systems. 1.2 Nonlinear Systems. 1.3 Equilibrium Points. 1.4 First-Order Autonomous Nonlinear Systems. 1.5 Second-Order Systems: Phase-Plane Analysis. 1.6 Phase-Plane Analysis of Linear Time-Invariant Systems. 1.7 Phase-Plane Analysis of Nonlinear Systems. 1.8 Higher-Order Systems. 1.9 Examples of Nonlinear Systems. 1.10 Exercises. Mathematical Preliminaries. 2.1 Sets. 2.2 Metric Spaces. 2.3 Vector Spaces. 2.4 Matrices. 2.5 Basic Topology. 2.6 Sequences. 2.7 Functions. 2.8 Differentiability. 2.9 Lipschitz Continuity. 2.10 Contraction Mapping. 2.11 Solution of Differential Equations. 2.12 Exercises. Lyapunov Stability I: Autonomous Systems. 3.1 Definitions. 3.2 Positive Definite Functions. 3.3 Stability Theorems. 3.4 Examples. 3.5 Asymptotic Stability in the Large. 3.6 Positive Definite Functions Revisited. 3.7 Construction of Lyapunov Functions. 3.8 The Invariance Principle. 3.9 Region of Attraction. 3.10 Analysis of Linear Time-Invariant Systems. 3.11 Instability. 3.12 Exercises. Lyapunov Stability II: Nonautonomous Systems. 4.1 Definitions. 4.2 Positive Definite Functions. 4.3 Stability Theorems. 4.4 Proof of the Stability Theorems. 4.5 Analysis of Linear Time-Varying Systems. 4.6 Perturbation Analysis. 4.7 Converse Theorems. 4.8 Discrete-Time Systems. 4.9 Discretization. 4.10 Stability of Discrete-Time Systems. 4.11 Exercises. Feedback Systems. 5.1 Basic Feedback Stabilization. 5.2 Integrator Backstepping. 5.3 Backstepping: More General Cases. 5.4 Examples. 5.5 Exercises. Input-Output Stability. 6.1 Function Spaces. 6.2 Input-Output Stability. 6.3 Linear Time-Invariant Systems. 6.4 Lp Gains for LTI Systems. 6.5 Closed Loop Input-Output Stability. 6.6 The Small Gain Theorem. 6.7 Loop Transformations. 6.8 The Circle Criterion. 6.9 Exercises. Input-to-State Stability. 7.1 Motivation. 7.2 Definitions. 7.3 Input-to-State Stability (ISS) Theorems. 7.4 Input-to-State Stability Revisited. 7.5 Cascade Connected Systems. 7.6 Exercises. Passivity. 8.1 Power and Energy: Passive Systems. 8.2 Definitions. 8.3 Interconnections of Passivity Systems. 8.4 Stability of Feedback Interconnections. 8.5 Passivity of Linear Time-Invariant Systems. 8.6 Strictly Positive Real Rational Functions. Exercises. Dissipativity. 9.1 Dissipative Systems. 9.2 Differentiable Storage Functions. 9.3 QSR Dissipativity. 9.4 Examples. 9.5 Available Storage. 9.6 Algebraic Condition for Dissipativity. 9.7 Stability of Dissipative Systems. 9.8 Feedback Interconnections. 9.9 Nonlinear L2 Gain. 9.10 Some Remarks about Control Design. 9.11 Nonlinear L2-Gain Control. 9.12 Exercises. Feedback Linearization. 10.1 Mathematical Tools. 10.2 Input-State Linearization. 10.3 Examples. 10.4 Conditions for Input-State Linearization. 10.5 Input-Output Linearization. 10.6 The Zero Dynamics. 10.7 Conditions for Input-Output Linearization. 10.8 Exercises. Nonlinear Observers. 11.1 Observers for Linear Time-Invariant Systems. 11.2 Nonlinear Observability. 11.3 Observers with Linear Error Dynamics. 11.4 Lipschitz Systems. 11.5 Nonlinear Separation Principle. Proofs. Bibliography. List of Figures. Index.

This paper is concerned with the mixed linear complementarity problem and the role it and its variants play in the stability analysis of the nonlinear complementarity problem and the Karush-Kuhn-Tucker system of a variational inequality problem. Under a nonsingular assumption, the mixed linear complementarity problem can be converted to the standard problem; in this case, the rich theory of the latter can be directly applied to the former. In this work, we employ degree theory to derive some sufficient conditions for the existence of a solution to the mixed linear complementarity problem in the absence of the nonsingularity property. Next, we extend this existence theory to the mixed nonlinear complementarity problem and establish a main stability result under a certain degree-theoretic assumption concerning the linearized problem. We then specialize this stability result and its consequences to the parametric variational inequality problem under the assumption of a unique set of multipliers. Finally, we consider the latter problem with the uniqueness assumption of the multipliers replaced by a convexity assumption and obtain stability results under some weak second-order conditions. In addition to the new existence results for the mixed linear complementarity problem, the main contributions of this paper in the stability category are the following: a resolution to a conjecture concerning the local solvability of a parametric variational inequality, the use of the generalized linear complementarity problem as a tool to broaden the second-order conditions, the characterization of the solution stability of the linear complementarity problem and the affine variational inequality problem in terms of the solution isolatedness under some weak hypotheses, and various stability theorems under some weak second-order conditions.

The Atiyah-Singer index theorem is a deep generalization of the classical Gauss- Bonnet theorem, including as special cases the Chern-Gauss-Bonnet theorem, the Hirzebruch signature theorem, and the Hirzebruch-Riemann-Roch theorem. Although the index theorem is about 35 years old at this point, it continues to have new applications in areas as apparently diverse as number theory and mathematical physics. The index theorem and its various generalizations (families index theorem, K-theoretic versions, etc.) admit many interpretations. We will choose the point of view that the index theorem expresses topological quantities in terms of geometric ones, just as in the Gauss-Bonnet theorem. This viewpoint leads to a heat equation proof of the index theorem, suggested by McKean and Singer [43] in the late 1960s and established by Gilkey [29], Patodi [55, 56], and Atiyah, Bott and Patodi [1] in the early 1970s. The heat equation method has since been refined by Getzler [28] (cf. [5]). In this chapter, we will give a complete heat equation proof for the Chern- Gauss-Bonnet theorem, and state without proof the Hirzebruch signature theorem, the Hirzebruch-Riemann-Roch theorem, and the Atiyah-Singer index theorem. Complete proofs can be found in [5] and [30]. We have also included a short introduction to characteristic classes. The Chern-Gauss Bonnet Theorem The key ideas in the heat equation method are (i) by Chapter 1, the long time behavior of the heat operator for the Laplacian on forms is controlled by the topology of the manifold in the form of the de Rham cohomology, (ii) the short time behavior is controlled by the geometry of the asymptotic expansion, as explained in Chapter 3, and (iii) certain combinations of heat operators will have time independent behavior.

This article addresses stability and stabilization of infinite delay systems, which are more general but also more challenging to deal with than bounded delay systems. Based on a general model of infinite delay systems and a newly approved key technical lemma, several new Lyapunov theorems for uniform asymptotical stability and exponential stability, respectively, are established. The stability results are more general than existing stability results, and the corresponding conditions are more easily satisfied than existing ones. Therefore, the new stability results are expected to be more widely applicable. These new Lyapunov theorems are then applied to the problems of stabilizing both time-invariant and time-varying linear systems with distributed infinite input delays, and the corresponding stabilizing controllers are developed. Key to obtaining the stabilizing controllers is to construct appropriate Lyapunov functionals for these systems via introduction of a regulation term into the Lyapunov functional. A distinctive advantage of the Lyapunov-based time-domain method proposed in this article over the existing frequency domain method is that the former can be adopted to deal with more general systems, such as time-varying linear systems or even nonlinear systems. Several examples are provided to illustrate the effectiveness of our results.

Modelling some diseases with large mortality rates worldwide, such as COVID-19 and cancer is crucial. Fractional differential equations are being extensively used in such modelling stages. However, exact analytical solutions for the solutions of such kind of equations are not reachable. Therefore, close exact solutions are of interests in many scientific investigations. The theory of stability in the sense of Ulam and Ulam–Hyers–Rassias provides such close exact solutions. So, this study presents stability results of some Caputo fractional differential equations in the sense of Ulam–Hyers, Ulam–Hyers–Rassias, and generalized Ulam–Hyers–Rassias. Two examples are introduced at the end to show the validity of our results. In this way, we generalize several recent interesting results.

A multivariable feedback system y¯(s) = G(s)x(s). x¯(s)=ū(s) − F(s)y(s) is considered, where G(s δ (gkl(s) is the n × n plant transfer matrix and f(s) = diag (f 1) [tdot] f 2(s) [tdot] f n(s)). By imposing some restrictions on the structure of G(s), we may use the eigenvalues λj(s), 1≤j≤n of G(s) to infer the stability of the closed-loop system in an arbitrary rectangular region Φ δ{F − ∞αk≤ƒkβk≤∞, 1≤k≤n}, which is a set of positive Lebesgue measure. The stability result contains Rosenbrock's stability theorem as a special case.

This paper is devoted to establish Bielecki–Ulam–Hyers–Rassias stability, generalized Bielecki–Ulam–Hyers–Rassias stability, and Bielecki–Ulam–Hyers stability on a compact interval [0,T], for a class of higher‐order nonlinear differential equations with fractional integrable impulses. The phrase ‘fractional integrable’ brings one to fractional calculus. Hence, applying usual methods for analysis offers many difficulties in proving the results of existence and uniqueness of solution and stability theorems. Picard operator is applied in showing existence and uniqueness of solution. Stability results are obtained by using the tools of fractional calculus and Hölder's inequality of integration. Along with tools of fractional calculus, Bielecki's normed Banach spaces are considered, which made the results more interesting. Copyright © 2017 John Wiley & Sons, Ltd.

Stability of motion is a central theme in the dynamics of mechanical systems. While the stability theory for systems with bilateral constraints is a well-established field, this monograph represents a systematic study of mechanical systems with unilateral constraints, such as unilateral contact, impact and friction. Such unilateral constraints give rise to non-smooth dynamical models for which stability theory is developed in this work. The book starts with the treatise of the mathematical background on non-smooth analysis, measure and integration theory and an introduction to the field of non-smooth dynamical systems. The unilateral constraints are modelled in the framework of set-valued force laws developed in the field of non-smooth mechanics. The embedding of these constitutive models in the dynamics of mechanical systems gives rises to dynamical models with impulsive phenomena. This book uses the mathematical framework of measure differential inclusions to formalise such models. The book proceeds with the presentation of stability results for measure differential inclusions. These stability results are then applied to nonlinear mechanical systems with unilateral constraints. The book closes with the study of the convergence property for a class of measure differential inclusions; a stability property for systems with time-varying inputs which is shown to be highly instrumental in the context of the control of mechanical systems with unilateral constraints. While the book presents a profound stability theory for mechanical systems with unilateral constraints, it also has a tutorial value on the modelling of such systems in the framework of measure differential inclusions. The work will be of interest to engineers, scientists and students working in the field of non-smooth mechanics and dynamics.

This paper studies fractional differential equations (FDEs) with mixed fractional derivatives. Existence, uniqueness, stability, and asymptotic results are derived.

Invariance and stability of fuzzy systems

We give a complete self-contained proof of Statman's finite completeness theorem and of a corollary of this theorem stating that the λ-definability conjecture implies the higher-order matching conjecture.The purpose of this note is to give a complete self-contained proof of Statman's finite completeness theorem and of a corollary of this theorem stating that the λ-definability conjecture implies the higherorder matching conjecture.Both results are proved in [8] (theorem 2 and 5).Although, since [8] assumes some familiarity with typed λ-calculus model theory and presents several results in short space, it may be not very accessible to readers not familiar with this subject.Section 1 gives the basic notations used in the paper.The reader not familiar with simply typed λ-calculus should consult Hindley and Seldin [3].Section 2 presents standard models for simply typed λcalculus, it is based on Henkin [2].Section 3 presents the Completeness theorem, it is based on Friedman [1], Plotkin [5] [6] and Statman [9].Section 4 presents the construction of a model for some equational theories.Section 5 presents Statman's finite completeness theorem.Both section 4 and 5 are based on [8].Section 6 presents the λ-definability conjecture.The notion of λ-definability is taken from Plotkin [5] [6].This conjecture has been studied by Plotkin and Statman.At last section 7 presents the higher order matching conjecture and the proof that the λ-definability conjecture implies the higher order matching conjecture.The decidability of higher order matching is conjectured in Huet [4], the equivalence of the higher order matching problem and the higher order matching problem with closed terms if proved in [7]

Borovik proposed an axiomatic treatment of Morley rank in groups, later modified by Poizat, who showed that in the context of groups the resulting notion of rank provides a characterization of groups of finite Morley rank [2]. (This result makes use of ideas of Lascar, which it encapsulates in a neat way.) These axioms form the basis of the algebraic treatment of groups of finite Morley rank undertaken in [1].There are, however, ranked structures, i.e., structures on which a Borovik-Poizat rank function is defined, which are not ℵ0-stable [1, p. 376]. In [2, p. 9] Poizat raised the issue of the relationship between this notion of rank and stability theory in the following terms: “… un groupe de Borovik est une structure stable, alors qu'un univers rangé n'a aucune raison de l'être …” (emphasis added). Nonetheless, we will prove the following:Theorem 1.1. A ranked structure is superstable.An example of a non-ℵ0-stable structure with Borovik-Poizat rank 2 is given in [1, p. 376]. Furthermore, it appears that this example can be modified in a straightforward way to give ℵ0-stable structures of Borovik-Poizat rank 2 in which the Morley rank is any countable ordinal (which would refute a claim of [1, p. 373, proof of C.4]). We have not checked the details. This does not leave much room for strenghthenings of our theorem. On the other hand, the proof of Theorem 1.1 does give a finite bound for the heights of certain trees of definable sets related to unsuperstability, as we will see in Section 5.

Journal Article Why the Edgeworth Process Assumption Isn't That Bad Get access Paul Madden Paul Madden University of Adelaide Search for other works by this author on: Oxford Academic Google Scholar The Review of Economic Studies, Volume 45, Issue 2, June 1978, Pages 279–283, https://doi.org/10.2307/2297343 Published: 01 June 1978 Article history Received: 01 January 1976 Accepted: 01 March 1977 Published: 01 June 1978

The purpose of this paper is to present a local theory of input/output stability of dynamical systems. It is shown that, by setting the problem in the context of extended binormed spaces it is possible to obtain local versions of some of the most important input/output stability results, such as the so-called small gain theorem and circle criterion for stability of nonlinear systems.

A slice theorem for the space of solutions of Einstein's equations