Robustness of nonuniform exponential trichotomies in Banach spaces

We establish the robustness under sufficiently small linear perturbations of nonuniform exponential trichotomies defined by linear equations x^′=A(t)x in Banach spaces. We also establish the continuous dependence on the perturbation of the constants in the notion of trichotomy. We consider both trichotomies in semi-infinite intervals and trichotomies in R.     

Robust nonuniform dichotomies and parameter dependence

We establish the robustness of linear cocycles with an exponential dichotomy, under sufficiently small Lipschitz perturbations, in the sense that the existence of an exponential dichotomy for a given cocycle persists under these perturbations. We consider cocycles in Banach spaces, as well as the general case of nonuniform exponential dichotomies, and also the general case of an exponential behavior ecρ(n), given by an arbitrary sequence ρ(n) including the usual exponential behavior ρ(n)=n as a very special case. Moreover, we show that the projections of the exponential dichotomies obtained from the perturbation vary continuously with the parameter, and in fact that they are locally Lipschitz on finite-dimensional parameters.     

Robustness of exponential dichotomies in mean

We consider the notion of an exponential dichotomy in mean, in which the exponential behavior in the classical notion of an exponential dichotomy is replaced by the much weaker requirement that the same happens in mean with respect to some probability measure. This includes as a special case any linear cocycle over a measure-preserving flow with nonzero Lyapunov exponents almost everywhere, such as the geodesic flow on a compact manifold of negative curvature. Our main aim is to show that the exponential behavior in mean is robust, in the sense that it persists under sufficiently small linear perturbations.     

Existence of nonuniform exponential dichotomies and a Fredholm alternative

We consider linear equations x^′=A(t)x in a Banach space, with nonuniform exponential dichotomies in and , and we give an admissibility condition for the existence of nonuniform exponential dichotomies in the whole line under sufficiently small perturbations. We also obtain a Fredholm alternative for the existence of bounded solutions under nonlinear perturbations.     

A Grobman-Hartman theorem for general nonuniform exponential dichotomies

For a nonautonomous dynamics with discrete time given by a sequence of linear operators Am, we establish a version of the Grobman-Hartman theorem in Banach spaces for a very general nonuniformly hyperbolic dynamics. More precisely, we consider a sequence of linear operators whose products exhibit stable and unstable behaviors with respect to arbitrary growth rates ecρ(n), determined by a sequence ρ(n). For all sufficiently small Lipschitz perturbations Am+fm we construct topological conjugacies between the dynamics defined by this sequence and the dynamics defined by the operators Am. We also show that all conjugacies are Hölder continuous. We note that the usual exponential behavior is included as a very special case when ρ(n)=n, but many other asymptotic behaviors are included such as the polynomial asymptotic behavior when ρ(n)=logn.     

Robustness analysis for parameter matrices of global exponential stable stochastic time varying delay systems

In this paper, we analyze the robustness of global exponential stable stochastic delayed systems subject to the uncertainty in parameter matrices. Given a globally exponentially stable systems, the problem to be addressed here is how much uncertainty in parameter matrices the systems can withstand to be globally exponentially stable. The upper bounds of the parameter uncertainty intensity are characterized by using transcendental equation for the systems to sustain global exponential stability. Moreover, we prove theoretically that, the globally exponentially stable systems, if additive uncertainties in parameter matrices are smaller than the upper bounds arrived at here, then the perturbed systems are guaranteed to also be globally exponentially stable. Two numerical examples are provided here to illustrate the theoretical results.     

Stable manifolds for nonuniform polynomial dichotomies

We establish the existence of smooth stable manifolds in Banach spaces for sufficiently small perturbations of a new type of dichotomy that we call nonuniform polynomial dichotomy. This new dichotomy is more restrictive in the “nonuniform part” but allow the “uniform part” to obey a polynomial law instead of an exponential (more restrictive) law. We consider two families of perturbations. For one of the families we obtain local Lipschitz stable manifolds and for the other family, assuming more restrictive conditions on the perturbations and its derivatives, we obtain C¹ global stable manifolds. Finally we present an example of a family of nonuniform polynomial dichotomies and apply our results to obtain stable manifolds for some perturbations of this family.     

Robustness of nonuniform exponential dichotomies in Banach spaces

We give conditions for the robustness of nonuniform exponential dichotomies in Banach spaces, in the sense that the existence of an exponential dichotomy for a given linear equation x^′=A(t)x persists under a sufficiently small linear perturbation. We also establish the continuous dependence with the perturbation of the constants in the notion of dichotomy and of the “angles” between the stable and unstable subspaces. Our proofs exhibit (implicitly) the exponential dichotomies of the perturbed equations in terms of fixed points of appropriate contractions. We emphasize that we do not need the notion of admissibility (of bounded nonlinear perturbations). We also obtain related robustness results in the case of nonuniform exponential contractions. In addition, we establish an appropriate version of robustness for nonautonomous dynamical systems with discrete time.     

NOTES ON GENERALIZED EXPONENTIAL DICHOTOMIES

ＮＯＴＥＳＯＮＧＥＮＥＲＡＬＩＺＥＤＥＸＰＯＮＥＮＴＩＡＬＤＩＣＨＯＴＯＭＩＥＳ￥ＺＨＡＮＧＷＥＩＮＩＡＮ（张伟年）（ＣｅｎｔｅｒｆｏｆＭａｔｈｅｍａｔｉｃａｌＳｃｉｅｎｃｅｓＣＩＣＡ，ｔｈｅＣｈｉｎｅｓｅＡｃａｄｅｍｙｏｆＳｃｉｅｎｃｅｓ，Ｃｈｅｎ...     

Stability of delay equations via Lyapunov functions

The importance of Lyapunov functions is well known. In the general setting of nonautonomous linear delay equations v^′=L(t)vt, we show how to characterize completely the existence of a nonuniform exponential contraction or of a nonuniform exponential dichotomy in terms of Lyapunov functions. This includes uniform exponential behavior as a very special case, and it provides an alternative (usually simpler and particularly more direct) approach to verify the existence of exponential behavior or to obtain the robustness of the dynamics under sufficiently small perturbations.     

Center manifolds for nonuniformly partially hyperbolic diffeomorphisms

We establish the existence of smooth invariant center manifolds for the nonuniformly partially hyperbolic trajectories of a diffeomorphism in a Banach space. This means that the differentials of the diffeomorphism along the trajectory admit a nonuniform exponential trichotomy. We also consider the more general case of sequences of diffeomorphisms, which corresponds to a nonautonomous dynamics with discrete time. In addition, we obtain an optimal regularity for the center manifolds: if the diffeomorphisms are of class C^k then the manifolds are also of class C^k. As a byproduct of our approach we obtain an exponential control not only for the trajectories on the center manifolds, but also for their derivatives up to order k.     

Smooth invariant manifolds in Banach spaces with nonuniform exponential dichotomy

We establish the existence of smooth stable manifolds for semiflows defined by ordinary differential equations v^′=A(t)v+f(t,v) in Banach spaces, assuming that the linear equation v^′=A(t)v admits a nonuniform exponential dichotomy. Our proof of the Ck smoothness of the manifolds uses a single fixed point problem in the unit ball of the space of Ck functions with α-Hölder continuous kth derivative. This is a closed subset of the space of continuous functions with the supremum norm, by an apparently not so well-known lemma of Henry (see Proposition 3). The estimates showing that the functions maintain the original bounds when transformed under the fixed-point operator are obtained through a careful application of the Faà di Bruno formula for the higher derivatives of the compositions (see (31) and (35)). As a consequence, we obtain in a direct manner not only the exponential decay of solutions along the stable manifolds but also of their derivatives up to order k when the vector field is of class Ck.     

Center manifolds under nonuniform hyperbolicity - maximal regularity

We establish the existence of (invariant) center manifolds with maximal Cr regularity for a nonautonomous dynamics with discrete time. We consider the general case of perturbations of a nonuniform exponential trichotomy. Our proof uses the fiber contraction principle and allows linear perturbations without any further effort.     

Robustness in multi-objective optimization using evolutionary algorithms

This work discusses robustness assessment during multi-objective optimization with a Multi-Objective Evolutionary Algorithm (MOEA) using a combination of two types of robustness measures. Expectation quantifies simultaneously fitness and robustness, while variance assesses the deviation of the original fitness in the neighborhood of the solution. Possible equations for each type are assessed via application to several benchmark problems and the selection of the most adequate is carried out. Diverse combinations of expectation and variance measures are then linked to a specific MOEA proposed by the authors, their selection being done on the basis of the results produced for various multi-objective benchmark problems. Finally, the combination preferred plus the same MOEA are used successfully to obtain the fittest and most robust Pareto optimal frontiers for a few more complex multi-criteria optimization problems.     

Robustness and exponentiality in redundant repairable systems

A redundant system of independent repairable components with general failure time and repair time distributions is considered. It is shown that the system failure time has an expectation dependent only on the mean failure times and mean repair times of the components and not on their distributions. An intuitive argument is provided, based on the time of onset of the quasi-stationary distribution on the set of working states, to show that for highly reliable systems, the system failure time is exponentially distributed to good approximation. The argument also justifies a simple figure of merit for the quality of the exponential approximation. A key ingredient in the analysis is the system relaxation time, an informal review of which is provided.     

Robustness for Inhomogeneous Poisson Point Processes

We consider robustness for estimation of parametric inhomogeneous Poisson point processes. We propose an influence functional to measure the effect of contamination on estimates. We also propose an M-estimator as an alternative to maximum likelihood estimator, show its consistency and asymptotic normality.     

Exponential dichotomies of impulsive dynamic systems with applications on time scales

In this paper, it is the first time that some important inequalities are obtained to estimate the exponential type of upper and lower bounds of solutions for the three representative classes of homogeneous impulsive dynamic systems on time scales. Based on these, some new criteria are established for admitting an exponential dichotomy of the impulsive dynamic systems. The obtained results are essentially new, even the time scale or . In addition, in applications, we apply the obtained results to discuss the almost periodic problems of a class of integro‐differential systems, and the numerical simulations are given to illustrate that our timescale methods are feasible and effective. Finally, we present the conclusion and further discussion related to this topic. Copyright © 2014 John Wiley & Sons, Ltd.     

Robustness analysis in Multi-Objective Mathematical Programming using Monte Carlo simulation

In most multi-objective optimization problems we aim at selecting the most preferred among the generated Pareto optimal solutions (a subjective selection among objectively determined solutions). In this paper we consider the robustness of the selected Pareto optimal solution in relation to perturbations within weights of the objective functions. For this task we design an integrated approach that can be used in multi-objective discrete and continuous problems using a combination of Monte Carlo simulation and optimization. In the proposed method we introduce measures of robustness for Pareto optimal solutions. In this way we can compare them according to their robustness, introducing one more characteristic for the Pareto optimal solution quality. In addition, especially in multi-objective discrete problems, we can detect the most robust Pareto optimal solution among neighboring ones. A computational experiment is designed in order to illustrate the method and its advantages. It is noteworthy that the Augmented Weighted Tchebycheff proved to be much more reliable than the conventional weighted sum method in discrete problems, due to the existence of unsupported Pareto optimal solutions.     

Robustness of Mann's algorithm for nonexpansive mappings

Mann's algorithm is proved robust in the sense that appropriately small perturbations do not alter the convergence of the algorithm. We prove this for nonexpansive mappings in a Banach space setting, which extends the initial work of Combettes [P.L. Combettes, On the numerical robustness of the parallel projection method in signal synthesis, IEEE Signal Process. Lett. 8 (2001) 45-47] where projections are considered in the Hilbert space framework.