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Publisher Summary This chapter presents the classical continuation theorem of Leray and Schauder, which has yielded a large number of existence results in the theory of nonlinear integral and differential equations. The continuation theory for multivalued A-proper and strongly A-closed mappings is developed in the chapter and applied to equations involving nonlinear perturbations of Fredholm mappings of index zero. New examples of A-proper and strongly A-closed mappings are presented, several general continuation theorems involving A-proper and strongly A-closed mappings are established, and applications of some of them to the (approximation) solvability of equations of the form f Є Ax + Nx involving certain positivity type conditions are given. The obtained results are constructive whenever the involved mappings are of A-proper type.

S.Klainerman, in [1] proved that for suficiently small nonlinear perturbations of linear dissipative equations and systems like: the wave equation, heat equation, linear isotropic elasticity equations etc., the corresponding solution of the initial-value problem for perturbed equations and system behave asymptotically like the solutions of the initial-value problem for linear unperturbed equations and system.

Part I deals with the problem of determining sufficient conditions under which the sum of two m-accretive operators on a closed convex set Q1 is m-accretive on Q1. Part II is concerned with the initial value problem: u′+Au+g(u)=v, u(0)=u0. Applications are given to the Boltzmann equation.

Let X be a Banach space and {A(t)|t e [0, T]} a family of closed linear, densely defined m-accretive operators in X. This paper is concerned with the additive perturbation of {A(t)|t e [0, T]} by a continuous family of nonlinear accretive operators {B(t)|t e [0, T]}. Namely solutions are provided for the integral equation u(t, τ, x) = W(t, τ)x −\(\mathop \smallint \limits_\tau ^t \)W(t, s)B(s) · · u(s, τ, x)ds, u(τ, τ, x) = x where W(t, s) is the linear evolution operator associated with the linear differential equation v'(t, s, x) + A(t)v(t, s, x) = 0, v(s, s, x) = x.

This short paper discusses the robustly decentralized stabilization of large-scale systems with nonlinear perturbations. The bounds are obtained for allowable nonlinear perturbations such that the resulting large-scale system with closed-loop subsystems remains stable.

Keywords: bifurcation ; nonlinear Hill's equation ; testfunctions Reference ANA-ARTICLE-1990-002View record in Web of Science Record created on 2008-12-10, modified on 2016-08-08

The sensitivity of the nonnegative solutions of the generalized continuous-time Riccati equations relative to perturbations in their coefficients is studied. A nonlocal nonlinear perturbation bound is obtained by the method of Lyapunov majorants.

An algorithm is presented to obtain a system which could observe n plants simultaneously, based on evolutionary strategies. We present a technique preserve an observer and its plants. We also give robust observability results, under nonlinear perturbations in the plant parameters, which are induced by substitutions of SPR functions in a given plant.

In this paper, the problem of the observer-based control for neutral systems with mixed time-varying delay and nonlinear perturbations is investigated. By using a class of novel augmented Lyapunov-Krasovskii functionals, delay dependent stability criteria are proposed to guarantee the stability of the observer-based control systems. In terms of linear matrix inequality techniques, the observer design and maximum allowable delay bound of neutral systems can be obtained. Finally, a numerical example is provided to illustrate the effectiveness of the proposed approach.

This paper presents improved stability results by introducing a new delay partitioning method based on the theory of geometric progression to deal with T–S fuzzy systems in the appearance of interval time-varying delays and nonlinear perturbations. A common ratio \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}α is applied to split the delay interval into multiple unequal subintervals. A modified Lyapunov–Krasovskii functional (LKF) is constructed with triple-integral terms and augmented factors including the length of every subintervals. In addition, the recently developed free-matrix-based integral inequality is employed to combine with the extended reciprocal convex combination and free weight matrices techniques for avoiding the overabundance of the enlargement when deducing the derivative of the LKF. Eventually, this developed research work can efficiently obtain the maximum upper bound of the time-varying delay with much less conservatism. Numerical results are conducted to illustrate the remarkable improvements of this proposed method.

We consider the Cauchy problem for systems of PDEs of the general form $$ u_t=P_0u + \eps_1P_1u +\eps_2 Q(u) + \sum_j D_j F_j(x,t), \quad u=u(x,t). $$ Here $P_0$ has constant coefficients. The terms $\eps_1 P_1u$ and $\eps_2 Q(u)$ describe linear and nonlinear perturbations, respectively, and $\textstyle\sum_j D_j F_j(x,t)$ is a forcing term, which decays to zero for $t\to \infty$. The perturbation terms are assumed to have conservation form. We call the system nonlinearly stable if the solution $u(x,t)$ with $u(x,0)=0$ remains smooth for all $t\geq 0$ and the maximum norm of u tends to zero for $t\to \infty$, provided that $\eps_1^2+\eps_2^2$ is sufficiently small. In the paper we give sufficient conditions for nonlinear stability. If the unperturbed system $u_t=P_0u$ is parabolic, then the Laplace transform technique is satisfactory to derive conditions for nonlinear stability. However, if $u_t=P_0u$ is hyperbolic or coupled parabolic-hyperbolic, then the Laplace transform technique fails if the perturb...

We investigate the impact of nonlinearity on the perfect imaging by segmentation in photonic lattices with disorder. We find the presence of strongly localized Anderson modes renders the imaging significantly more susceptible to nonlinear perturbations.

Logarithmic-in-time slow dynamics has been found for individual cracks in a solid. Furthermore, this phenomenon is observed during both the crack acoustic conditioning and the subsequent relaxation. A thermoelastic mechanism is suggested which relates the log-time behavior to the essentially 2D character of the heating and cooling of the crack perimeter and inner contacts. Nonlinear perturbation of the contacts by a stronger (pump) wave causes either softening or hardening of the sample, and induces either additional absorption or transparency for a weaker (probe) acoustic wave depending on frequency of the latter.

Asymptotically stable linear systems subject to delayed time-varying and nonlinear perturbations are considered. Razumikhin-type theorems are used to obtain easy-to-compute bounds on the perturbations so that the systems remain stable. Results indicate that if delayed perturbations are included, then the bound is reduced as compared to the one for nondelayed perturbations. However, in certain cases previously obtained bounds for the nondelayed perturbations guarantee stability even when delayed perturbations are in effect. >