Stability of Discrete-Time Regime-Switching Dynamic Systems with Delays

Abstract

This work focuses on stability of regime-switching discrete-time systems with delays. Two-time-scale formulation is used for the purpose of reduction of complexity. It is demonstrated that associated with the original system, there is a limit system that is a switching diffusion process. An interesting problem is concerned with if the stability of the limit switching diffusion process can be carried over to the original system. This question is answered in the article. Furthermore, path excursion, mean recurrence time, and the associated error bounds are considered.     

A GEOMETRIC SPLITTING AND THE C∞ STABILITY OF MAPS FROM SEMIANALYTIC SETS

Abstract

John Mather has proved that infinitesimal stability implies stability for proper maps in the category of smooth manifolds. This result gives a computable algebraic criterion for stability. In this paper it is shown that there is an extension of Mather's result when the range is only assumed to be a compact semianalytic set of some real Euclidean space—this class of spaces is an obvious maximal candidate for which computations can be carried out using only classical polynomial algebra. The proof depends on a splitting theorem for the restriction map from the smooth functions on a Euclidean space to those on a closed subset and is proved by an algebraic-geometric method derived from the work of B. Malgrange. No knowledge of functional analysis is assumed although an alternative analytic method for proving the main result is also indicated. Only simple applications are given (mostly to functions defined locally in the neighbourhood of an isolated hypersurface singularity of the type studied by J. Milnor and others) since the author intends to publish a fairly comprehensive study of stability (smooth and C°) of smooth maps on closed semianalytic sets.     

Deterministic and stochastic equations of motion arising in Oldroyd fluids of order one: existence,uniqueness, exponential stability and invariant measures

Abstract

In this work, we consider the two-dimensional viscoelastic fluid flow equations, arising from the Oldroyd model for the non-Newtonian fluid flows. We investigate the well-posedness of such models in two-dimensional bounded and unbounded (Poincaré domains) domains, both in deterministic and stochastic settings. The existence and uniqueness of weak solution in the deterministic case is proved via a local monotonicity property of the linear and nonlinear operators and a localized version of the Minty-Browder technique. Some results on the exponential stability of stationary solutions are also established. The global solvability results for the stochastic counterpart are obtained by a stochastic generalization of the Minty-Browder technique. The exponential stability results in the mean square as well as in the pathwise (almost sure) sense are also discussed. Using the exponential stability results, we finally prove the existence of a unique invariant measure, which is ergodic and strongly mixing.     

Sur les Suranneaux Minimaux. II

Abstract

This work includes a continuation of the survey that has been made on the minimal overrings by Oukessou and Miri (Oukessou, M., Miri, A. (1999). Sur les suranneaux minimaux. Extracta Mathematicae 14(3):333–347), we are going to examine the stability of the structures of pseudo valuation domain, S-Domain and catenarian ring. We will start this work with establishing some properties of the minimal overrings.     

A Dynamical System for Plant Pattern Formation: A Rigorous Analysis

Summary. We present a rigorous mathematical analysis of a discrete dynamical system modeling plant pattern formation. In this model, based on the work of physicists Douady and Couder, fixed points are the spiral or helical lattices often occurring in plants. The frequent occurrence of the Fibonacci sequence in the number of visible spirals is explained by the stability of the fixed points in this system, as well as by the structure of their bifurcation diagram. We provide a detailed study of this diagram.     

Existence and stability in the α-norm for partial functional differential equations of neutral type

In this work, we study a general class of partial neutral functional differential equations. We assume that the linear part generates an analytic semigroup and the nonlinear part is Lipschitz continuous with respect to the é-norm associated to the linear part. We discuss the existence, uniqueness, regularity and stability of solutions. Our results are illustrated by an example. This work extends previous results on partial functional differential equations (Fitzgibbon and Parrot, Nonlinear Anal., TMA 16, 479–487 (1991), Hale, Rev. Roum. Math. Pures Appl. 39, 339–344 (1994), Hale, Resen. Inst. Mat. Estat. Univ. Sao Paulo 1, 441–457 (1994), Travis and Webb, Trans. Am. Math. Soc. 240 129–143 (1978), Wu and Xia, J. Differ. Equ. 124 247–278 (1996)). Mathematics Subject Classification (1991) 34K20, 34K30, 34K40, 47D06     

On the two-dimensional tidal dynamics system: stationary solution and stability

ABSTRACT

In this work, we consider the two-dimensional stationary and non-stationary tidal dynamic equations and examine the asymptotic behavior of the stationary solution. We prove the existence and uniqueness of weak and strong solutions of the stationary tidal dynamic equations in bounded domains using compactness arguments. Using maximal monotonicity property of the linear and nonlinear operators, we also establish that the solvability results are even valid in unbounded domains. Later, we obtain a uniform Lyapunov stability of the steady state solution. Finally, we remark that the stationary solution is exponentially stable if we add a suitable dissipative term in the equation corresponding to the deviations of free surface with respect to the ocean bottom. This exponential stability helps us to ensure the mass conservation of the modified system, if we choose the initial data of the modified system as stationary solution.     

Linearity of stability conditions

Abstract

For modules over an artin algebra, a linear stability condition is given by a “central charge” and a nonlinear stability condition is given by the wall-crossing sequence of a “green path.” Finite Harder-Narasimhan stratifications of the module category, maximal forward hom-orthogonal sequences and maximal green sequences, defined using Fomin-Zelevinsky quiver mutation are shown to be equivalent to finite nonlinear stability conditions when the algebra is hereditary. This is the first of a series of three papers whose purpose is to determine all maximal green sequences of maximal length for quivers of affine type A and determine which are linear.     

Stability analysis of thermosolutal second-order fluid in porous Bénard layer

Abstract Linear and nonlinear stability of the motionless state of thermosolutal second-order fluid in porous Bénard layer is investigated via Lyapunov direct method on the basis of Brinkman’s modification of the Darcy’s model. Critical Rayleigh numbers for linear and nonlinear stability are obtained for free-free, rigid-rigid and rigid-free boundaries. The stabilizing effect of solute concentration and the destabilizing effect of medium permeability and porosity on the basic motion are proved. In particular, for certain range of system parameters the sufficient and necessary condition for stability coincide. Keywords: Second-order fluid, Nonlinear stability, Porous medium, Solute concentration Mathematics Subject Classification (2000): 35Q35, 46N20, 76E06     

Resolvent Estimates and Quantification of Nonlinear Stability

Abstract The aim of this paper is to clarify the role played by resolvent estimates for nonlinear stability. We will give examples showing that a large resolvent may lead to a small domain of nonlinear stability. In other examples the resolvent is large, but the domain of nonlinear stability is completely unrestricted. Which case prevails depends on the details of the problem. We will also show that the size of the resolvent depends in an essential way on the norms that are used. * Supported by Office of Naval Research n00014 90 j 1382 ** Supported by NSF Grant DMS-9404124 and DOE Grant DE-FG03-95ER25235     

Spectrally Determined Growth for Creeping Flow of the Upper Convected Maxwell Fluid

Abstract. While it is well known that the stability of Newtonian flows is determined by the eigenvalues of a linearized equation, there are no general results of this type for non-Newtonian fluids. In this paper, we show that linear stability of steady creeping flows of the upper convected Maxwell fluid is indeed determined by the spectrum of the linearized operator. The proof uses the theory of evolution semigroups over dynamical systems.     

Existence and Hyers-Ulam stability of random impulsive stochastic functional differential equations with finite delays

ABSTRACT

In this paper, we investigate the existence and Hyers-Ulam stability for random impulsive stochastic functional differential equations with finite delays. Firstly, we prove the existence of mild solutions to the equations by using Krasnoselskii's fixed point. Then, we investigate the Hyers-Ulam stability results under the Lipschitz condition on a bounded and closed interval. Finally, an example is given to illustrate our results.     

Delay-Dependent Criteria for Robust Stabilization of Markovian Switching Networks with Time-Varying Delay

Abstract

A problem of feedback stabilization of hybrid systems with time-varying delay and Markovian switching is considered. Delay-dependent sufficient conditions for stability based on linear matrix inequalities (LMI's) for stochastic asymptotic stability is obtained. The stability result depended on the mode of the system and of delay-dependent. The robustness results of such stability concept against all admissible uncertainties are also investigated. This new delay-dependent stability criteria is less conservative than the existing delay-independent stability conditions. An example is given to demonstrate the obtained results.     

MPCC: on necessary conditions for the strong stability of C-stationary points

ABSTRACT

We consider the concept of strongly stable C-stationary points for mathematical programs with complementarity constraints. The original concept of strong stability was introduced by Kojima for standard optimization programs. Adapted to our context, it refers to the local existence and uniqueness of a C-stationary point for each sufficiently small perturbed problem. The goal of this paper is to discuss a Mangasarian-Fromovitz-type constraint qualification and, mainly, provide two conditions which are necessary for strong stability; one is another constraint qualification and the second one refers to bounds on the number of active constraints at the point under consideration.     

Novel trend of mixed Minkowski volumes applications

ABSTRACT

In this paper, a family of equations with uncertain values of parameters is investigated. An application of mixed Minkovski volumes is proposed, and conditions for stability, asymptotic stability and instability of the stationary solutions are established.     

On the Free Boundary Conditions for a Dynamic Shell Model Based on Intrinsic Differential Geometry

The mathematical theory behind the modeling of shells is a crucial issue in many engineering problems. Here, the authors derive the free boundary conditions and associated strong form of a dynamic shallow Kirchhoff shell model based on the intrinsic geometry methods of Michael Delfour and Jean-Paul Zolésio. This model relies on the oriented distance function which describes the geometry. This is an extension of the work done in [J. Cagnol, I. Lasiecka, C. Lebiedzik and J.-P. Zolésio (2002). Uniform stability in structural acoustic models with flexible curved walls. J. Differential Equation, 186(1), 88–121.], where the model was derived for clamped boundary conditions only. In the current article, manipulations with the model result in a cleaner form where the displacement of the shell and shell boundary is written explicitly in terms of standard tangential operators.     

Stability Analysis of nth Order Nonlinear Impulsive Differential Equations in Quasi–Banach Space

Abstract

This article is about Ulam’s type stability of n^th order nonlinear differential equations with fractional integrable impulses. It is a best procession to the stability of higher order fractional integrable impulsive differential equations in quasi–normed Banach space. Different Ulam’s type stability results are obtained by using the definitions of Riemann–Liouville fractional integral, Hölder’s inequality and the beta integral inequality.     

Stability in Terms of Two Measures for Stochastic Differential Equations with Markovian Switching

Abstract

In this paper, we investigate the stability in terms of two measures for stochastic differential equations with Markovian switching by using the method of Lyapunov functions. Our new theory can not only be used to show a given system to be stochastically stable in the classical sense, but can also be used to deal with some situations where the classical stability theory is not applicable.     

The Stability of Self-Organized Rule-Following Work Teams

Self-organized rule-following systems are increasingly relevant objects of study in organization theory due to such systems&2018; capacity to maintain control while enabling decentralization of authority. This paper proposes a network model for such systems and examines the stability of the networks&2018; repetitive behavior. The networks examined are Ashby nets, a fundamental class of binary systems: connected aggregates of nodes that individually compute an interaction rule, a binary function of their three inputs. The nodes, which we interpret as workers in a work team, have two network inputs and one self-input. All workers in a given team follow the same interaction rule.We operationalize the notion of stability of the team&2018;s work routine and determine stability under small perturbations for all possible rules these teams can follow. To study the organizational concomitants of stability, we characterize the rules by their memory, fluency, homogeneity, and autonomy. We relate these measures to work routine stability, and find that stability in ten member teams is enhanced by rules that have low memory, high homogeneity, and low autonomy.     

Stability in Probability of Nonlinear Stochastic Volterra Difference Equations with Continuous Variable

Abstract

The general method of Lyapunov functionals construction, that was proposed by Kolmanovskii and Shaikhet and successfully used already for functional-differential equations, difference equations with discrete time, difference equations with continuous time, and is used here to investigate the stability in probability of nonlinear stochastic Volterra difference equations with continuous time. It is shown that the investigation of the stability in probability of nonlinear stochastic difference equation with order of nonlinearity more than one can be reduced to investigation of the asymptotic mean square stability of the linear part of this equation.