English

In the present chapter nonlinear neutral type functional differential systems with separated linear parts and continuous nonlinearities are considered. In Section 12.1 we introduce the stability definitions for nonlinear neutral type systems. Absolute stability is discussed in Sections 12.2 and 12.3. Section 12.4 deals with the stability in the first approximation.

In this paper, the weak approximate controllability of neutral stochastic evolution differential systems with poisson jump is considered. When these linear systems is approximate controllability and evolution operator is not compact, the sufficient conditions for the weak approximate controllability of these associated nonlinear stochastic systems in Hilbert spaces is obtained by using contracting fixed-point theorem.

This study investigates the stability and existence of periodic solutions in neutral differential systems with time delays and variable coefficients. Utilizing Krasnoselskii’s fixed point theorem, we demonstrate a set of sufficient conditions ensuring the existences of such a periodic solution. This involves transforming the system into an equivalent integral form before applying the fundamental matrix solutions alongside Floquet theory. In addition, we will analyze the asymptotic stability of these solutions, thus establishing new conditions that can ensure stability. The practical relevance of our theoretical results is supported through numerical examples, validating the proposed approach, and highlighting its suitability in areas such as electrical circuits, control systems, and biological modeling. This study extends previous work and thereby offers a detailed framework intended for use in studying neutral differential systems where time delays are present.

Neutral stochastic dynamic systems have received broad attention, because the problems are not only great theoretical significance, but also have played an important role in many ways such as engineering control, finance, communication equipment, social science, etc. In this paper, we consider stability of neutral stochastic partial differential dynamic systems with Markovian switching. Some new sufficient conditions proving the stability in mean square of this neutral stochastic dynamic systems are derived by employing the fixed point approach. Some well-known results are improved and generalized.

Robust stability of uncertain neutral linear stochastic differential delay system

Consider the nonlinear neutral functional differential inclusion (i) $$(d/dt)D(t, x_t ) \in R(t, x_t )$$ , whereD is a continuous operator onIXC, linear inx t , indeed of the form (4) below, with kernelD(t, ·)={0}, and atomic at 0, andR is nonempty, closed, and convex. Here,I≡[t 0,t I ] andC=C([-h,0],E n ). In (i), the derivative is specified in terms of the state at timet as well as the state and the derivative of the state for values oft precedingt. We use the Fan fixed-point theorem to prove the existence of a solution of (i) which satisfies two-point boundary values $$x_\omega = \phi _0 ,x_{t_1 } = \phi _1$$ , where φ0, φ1 belong toC. We next apply this existence result to study the exact function space controllability of the neutral functional differential system (ii) $$(d/dt)D(t, x_t ) = f(t, x_t , u), u(t) \in \Omega (t, x_t )$$ . We present sufficient conditions onf and Ω which imply exact controllability between two fixed functions inC.

ABSTRACTIn this paper, we consider a class of neutral stochastic differential systems with variable delays and give sufficient conditions to ensure that the zero solution is asymptotically stable in the mean square by means of a generalisation of Banach's contraction principle. These conditions do not require the boundedness of delays, nor do they ask for a fixed sign on the coefficient functions. The results are shown to improve the previous globally stable results derived in the literature.

Easily testable necessary and sufficient algebraic criteria for delay-independent stability of a class of neutral differential systems

In this paper, we investigate the existence of a set with 2kT-periodic solutions for n-dimensional p-Laplacian neutral differential systems with a time-varying delay (varphi_{p}(u(t)-Cu(t-tau ))')'+ frac{d}{dt}nabla F(u(t))+G(u(t-gamma (t)))=e_{k}(t) based on the coincidence degree theory of Mawhin. Combining this with the conclusion about uniform convergence and limit, we obtain the corresponding results on the existence of homoclinic solutions.

In this paper, we discuss a class of impulsive evolution neutral functional differential inclusions with infinite delay. The existence of mild solutions of these evolution neutral differential systems is obtained by using the new nonlinear alternatives of Leray-Schauder type.

Download options Please wait... Article information DOI https://doi.org/10.1039/P19930002969 Article type Paper Download Citation J. Chem. Soc., Perkin Trans. 1, 1993, 2969-2973 BibTex EndNote MEDLINE ProCite ReferenceManager RefWorks RIS Permissions Request permissions Chiral chlorobismuthines stabilized by the intramolecular coordination of an N,N-dimethylamino group: X-ray structure analysis, asymmetric induction at the bismuth centre, and dynamic behaviour in solution H. Suzuki, T. Murafuji, Y. Matano and N. Azuma, J. Chem. Soc., Perkin Trans. 1, 1993, 2969 DOI: 10.1039/P19930002969 To request permission to reproduce material from this article, please go to the Copyright Clearance Center request page. If you are an author contributing to an RSC publication, you do not need to request permission provided correct acknowledgement is given. If you are the author of this article, you do not need to request permission to reproduce figures and diagrams provided correct acknowledgement is given. If you want to reproduce the whole article in a third-party publication (excluding your thesis/dissertation for which permission is not required) please go to the Copyright Clearance Center request page. Read more about how to correctly acknowledge RSC content. Search articles by author Hitomi Suzuki Toshihiro Murafuji Yoshihiro Matano Nagao Azuma

From the following studies, Artstein and Tradma (1982),

This paper is aimed at establishing sufficient computable criteria for the Euclidean null controllability of an infinite neutral differential system, when the controls are essentially bounded measurable functions on finite intervals, with values in a compact subset U of an m-dimensional Euclidean space with zero in its interior. Our results are obtained by exploiting the stability of the free system and the rank criterion for properness of the controlled system. An example is also given.

In this paper, a fractional order neutral differential equation with deviated argument is studied. The existence and uniqueness of mild solution are obtained by using Banach contraction principle without imposing additional assumptions such as compactness, analyticity, uniform boundedness etc. We also discuss the approximate controllability of a fractional neutral differential equation with deviated argument by assuming a simple range condition \(({ HR})\). Thereby, we remove the need to assume the invertibility of a controllability operator used by Tai (Appl. Math. Lett. 24: 2158–2161, 2011), which fails to exist in infinite dimensional spaces if the associated semigroup is compact. Our approach also removes the need to check the invertibility of the controllability Gramian operator and associated limit condition used by Sakthivel et al. (Rep. Math. Phys. 70: 291–311, 2012), which are practically difficult to verify and apply. An example is provided to illustrate the presented theory.

The problems of determining the global asymptotic stability and global exponential stability for a class of norm-bounded nonlinear neutral differential systems with constant or time-varying delays are investigated in this work. Lyapunov method was used to derive some useful criteria of the systems’ global asymptotic stability and global exponential stability. The stability conditions are formulated as linear matrix inequalities (LMIs) which can be easily solved by various convex optimization algorithms. Moreover, for the exponentially stable system, the exponential convergence rates of the system’s states can be estimated by some parameters of the LMIs. Numerical examples are given to illustrate the application of the proposed method.