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Positivity and stability of linear functional differential equations with infinite delay
Abstract
General linear functional differential equations with infinite delay are considered. We first give an explicit criterion for positivity of the solution semigroup of linear functional differential equations with infinite delay and then a Perron-Frobenius type theorem for positive equations. Next, a novel criterion for the exponential asymptotic stability of positive equations is presented. Furthermore, two sufficient conditions for the exponential asymptotic stability of positive equations subjected to structured perturbations and affine perturbations are provided. Finally, we applied the obtained results to problems of the exponential asymptotic stability of Volterra integrodifferential equations. To the best of our knowledge, most of the results of this paper are new.
... Recently, problems of differential equations with infinite delay have attracted much attention from researchers, see e.g., [1,2,[25][26][27][28][29][30][31][32][33][34]44,45,47,48]. In particular, motivated by many applications in various scientific areas, especially in cellular neural networks, Cohen-Grossberg neural networks, bidirectional associative memory (BAM) neural networks model, problems of stability of differential equations with infinite delay have been studied intensively, see e.g., [1,2,20,[26][27][28][29][30][31][32][33]39,44,47,50]. ...
... Recently, problems of differential equations with infinite delay have attracted much attention from researchers, see e.g., [1,2,[25][26][27][28][29][30][31][32][33][34]44,45,47,48]. In particular, motivated by many applications in various scientific areas, especially in cellular neural networks, Cohen-Grossberg neural networks, bidirectional associative memory (BAM) neural networks model, problems of stability of differential equations with infinite delay have been studied intensively, see e.g., [1,2,20,[26][27][28][29][30][31][32][33]39,44,47,50]. ...
... In particular, if A 0 (·) ≡ A 0 ∈ R n×n and A k (·) ≡ A k ∈ R n×n , for each k ∈ N and B(t, s) := C(s), t ∈ R, s ∈ R − then (1) reduces to (2). Assume that ∞ k=1 e γ h k A k < ∞ and 0 −∞ e −γ s C(s) ds < ∞, for given γ > 0. Then (2) is a positive system if and only if A 0 ∈ R n×n is a Metzler matrix, A k ∈ R n×n + , for each k ∈ N and C(s) ∈ R n×n + for all s ∈ R − , see [26,31] for further information. Suppose (2) is positive. ...
Time-varying differential systems with infinite delay are considered. Explicit criteria for global exponential stability of linear (nonlinear) systems are presented. Furthermore, an explicit robust stability bound for linear systems subject to time-varying perturbations is given. The exponential stability criteria for nonlinear systems are used to investigate exponential stability of equilibria of neural networks. Three examples are given to illustrate obtained results. To the best of our knowledge, the results of this paper are new.
... In Theorems 3.3 and 3.5, if the ϱ(t) has the form of a rational polynomial function (for example, ϱ(t) = 1 t k +1 for t ≥ 0) then we obtain the polynomial stability condition for general linear non-autonomous functional differential equations with infinite delay. Moreover, if ϱ(t) = e − t 0 ω(s)ds or ϱ(t) = e −γt then we obtain some results the exponential stability of linear non-autonomous FDEs in [12,32,34,35]. ...
... The following corollary has been known in [11,32]. For convenient reader, here we give a proof used Corollary 3.9 and Theorem 2.1. ...
... It is easy to see that the asymptotically leading terms in (27) are ...
It is well known that the linear differential equation with continuous delay , , , and has a positive solution on if an explicit criterion of the integral type
\begin{equation*}
\int_{t-\tau (t)}^{t}p(s){\mathrm d}s\leq \frac{1}{\mathrm{e}}
\end{equation*}
holds for all . In this paper new integral explicit criteria, which essentially supplement related results in the literature are established. For example, if, for and a fixed , the integral inequality
\begin{equation*}
\int_{t-\tau /2}^{t}p(s)\,{\mathrm d}s\leq \frac{1}{2\mathrm{e}}+\frac{\mu \tau
^{3}}{96t^{3}\mathrm{e}}
\end{equation*}
holds, then there exists a and a positive solution x=x(t) on . Examples illustrating the effectiveness of the results are given.
Recently, authors [2] have discussed some equivalent relations for-uniform stabil-ities of a given equation and those of its limiting equations by using the skew product ow constructed by quasi-processes on a general metric space. In 1992, Murakami and Yoshizawa [6] pointed out that for functional di erential equations with in nite delay on a fading memory space B = B(( 1; 0]; R,)-stability is a useful tool in the study of the existence of almost periodic solutions for almost periodic systems and they proved that-total stability is equivalent to BC-total stability. The purpose of this paper is to show that equivalent relations established by Mu-rakami and Yoshizawa [6] holds even for functional di erential equations with in nite delay on a fading memory space B = B(( 1; 0]; X ) with a general Banach space X.
All processes take time to complete. While physical processes such as acceleration and deceleration may not take much time compare to the times needed to travel most distances, biological process times such as gestation periods and maturation times can be substantial when compared to the data collection times in most population studies. Therefore, it is often imperative to explicitly incorporate these process times in mathematical models of population dynamics. These process times are often called delay times and the models that incorporate such delay times are referred as delay differential equation (DDE) models. Recent theoretical and computational advancements in delay differential equations reveal that DDEs are capable of generating rich and plausible dynamics with realistic parameter values. Naturally occurring complex dynamics are often naturally generated by well formulated DDE models. This is simply due to the fact that a DDE operates on an infinite dimensional space consisting of continuous functions which accommodates high dimensional dynamics. For example, the Lotka-Volterra predator-prey model with crowding effect does not produce sustainable oscillatory solutions that describe population cycles, yet the Nicholson's blowflies model can generate rich and complex dynamics.
This paper deals with a class of partial functional di¤erential equations with infinite delay. Supposing that the linear part is a Hille-Yosida operator but not necessarily densely defined, and employing the integrated semigroups and dissipative dynamics theory, we present some appropriate conditions to guarantee existence of a global attractor.
We first give a criterion for positivity of the solution semigroup of linear Volterra integro-differential systems. Then,
we offer some explicit conditions under which the solution of a positive linear Volterra system is exponentially stable or
(robustly) lies in L2[0,+∞).
In this note a simple formula for the real stability radius of uncertain positive linear continuous time systems is established and it is shown that the real stability radius coincides with the complex one. Arbitrary disturbance norms induced by monotonic vector norms (e g. p-norms, 1 ≤ p ≤ ∞) are considered. The distance of intervals of positive systems from instability is also determined.
Nonnegative and compartmental dynamical system models are derived from mass and energy balance considerations and involve the exchange of nonnegative quantities between subsystems or compartments. These models are widespread in biological and physical sciences and play a key role in understanding these processes. A key physical limitation of such systems is that transfers between compartments is not instantaneous and realistic models for capturing the dynamics of such systems should account for material in transit between compartments. In this paper, we present necessary and sufficient conditions for stability of nonnegative and compartmental dynamical systems with time delay. Specifically, asymptotic stability conditions for linear and nonlinear nonnegative dynamical systems with time delay are established using linear Lyapunov–Krasovskii functionals.
Nonnegative dynamical system models are derived from mass and energy balance considerations that involve dynamic states whose values are nonnegative. These models are widespread in biological, physiological, and ecological sciences and play a key role in the understanding of these processes. In this paper we develop several results on stability, dissipativity, and stability of feedback interconnections of linear and nonlinear nonnegative dynamical systems. Specifically, using linear Lyapunov functions we develop necessary and sufficient conditions for Lyapunov stability, semistability, that is, system trajectory convergence to Lyapunov stable equilibrium points, and asymptotic stability for nonnegative dynamical systems. In addition, using linear and nonlinear storage functions with linear supply rates we develop new notions of dissipativity theory for nonnegative dynamical systems. Finally, these results are used to develop general stability criteria for feedback interconnections of nonnegative dynamical systems.
We study stability radii of linear retarded systems described by general linear functional differential equations. A lower and an upper bound for the complex stability radius with respect to multi-perturbations are given. Furthermore, in some special cases concerning the structure matrices, the complex stability radius can precisely be computed via the associated transfer function. Then, the class of positive linear retarded systems is studied in detail. It is shown that for this class, complex, real and positive stability radius under multi-perturbations or multi-affine perturbations coincide and can be computed by simple formulae expressed in terms of the system matrices.
This paper deals with a particular class of positive systems. The state components of a positive system are positive or zero for all positive times. These systems are often encountered in applied areas such as chemical engineering or biology. It is shown that for this particular class the first orthant contains an invariant ray in its interior. An invariant ray generalizes the concept of an eigenvector of linear systems to nonlinear homogeneous systems. Then sufficient conditions for uniqueness of this ray are given. The main result states that the vector field on an invariant ray determines the stability properties of the zero solution with respect to initial conditions in the first orthant. The asymptotic behavior of the solutions is examined. Finally, we compare our results to the Perron-Frobenius theorem, which gives a detailed picture of the dynamical behavior of positive linear systems.
So far there are not many results on the stability for stochastic functional
differential equations with infinite delay. The main aim of this paper is to establish some new
criteria on the stability with general decay rate for stochastic functional differential equations
with infinite delay. To illustrate the applications of our theories clearly, this paper also
examines a scalar infinite delay stochastic functional differential equations with polynomial
coefficients.
If A A is a nonnegative matrix whose associated directed graph is strongly connected, the Perron-Frobenius theorem asserts that A A has an eigenvector in the positive cone, ( R + ) n (\mathbb R^{+})^n . We associate a directed graph to any homogeneous, monotone function, f : ( R + ) n → ( R + ) n f: (\mathbb R^{+})^n \rightarrow (\mathbb R^{+})^n , and show that if the graph is strongly connected, then f f has a (nonlinear) eigenvector in ( R + ) n (\mathbb R^{+})^n . Several results in the literature emerge as corollaries. Our methods show that the Perron-Frobenius theorem is “really” about the boundedness of invariant subsets in the Hilbert projective metric. They lead to further existence results and open problems.
In this paper, we give a characterization of spectral abscissa of positive linear functional differential equations. Then
the obtained result is applied to derive necessary and sufficient conditions for the exponential stability of positive linear
functional differential equations. Finally, we give an extension of the classical Perron–Frobenius theorem to positive linear
functional differential equations.
The present monograph (by 4 authors!) is apparently an important and very interesting presentation of the abstract Cauchy problem treated especially by means of the (vector-valued) Laplace and Laplace-Stieltjes transforms. It appears as a worthy continuation of the classical books by it Hille [Am. Math. Soc. (1948; Zbl 0033.06501)] and it Hille and it Phillips [Am. Math. Soc. (1957; Zbl 0078.10004)] about functional analysis and semigroups.par The basic idea reads: If A is a closed linear operator on a Banach space X, one considers the Cauchy problem ("initial value problem") where is given. If u(cdot) is an exponentially bounded continuous function, which is also a (mild) solution, that is: and if one considers the Laplace transform: which converges for large lambda, then (lambda large), and conversely. Thus, if -- the resolvent set of A -- then .par This fundamental relationship indicates that the Laplace transform is the link between solutions and resolvents, between Cauchy problems and spectral properties of operators.par A further study concerns criteria to decide whether a given function is a Laplace transform. Such results -- in the vector-valued case -- when applied to the resolvent of an operator, would give information on the solvability of the Cauchy problem.par Finally, let us note that our aim here is not to provide a summary -- or appreciation -- of the wealth of concepts contained in this 500 pages book. We invite all interested mathematicians to study this monograph, or any part of it. The reward should be considerable, as always is when reading great mathematics.
Exponential asymptotic stability of the zero solution of the scalar linear Volterra equation is studied. Roughly speaking, it is proved that the exponential asymptotic stability can be characterized by a growth condition on B(t). The result answers the problem posed by Corduneanu and Lakshmikantham.
Linear differential equations in Banach spaces are systematically treated with the help of Laplace transforms. The central
tool is an “integrated version” of Widder’s theorem (characterizing Laplace transforms of bounded functions). It holds in
any Banach space (whereas the vector-valued version of Widder’s theorem itself holds if and only if the Banach space has the
Radon-Nikodým property). The Hille-Yosida theorem and other generation theorems are immediate consequences. The method presented
here can be applied to operators whose domains are not dense.
In this paper, we study a class of partial functional differential equations with finite delay, whose linear part is not necessarily densely defined but satisfies the Hille–Yosida condition. Using the classical theory about global attractors in infinite dimensional dynamical systems, we establish some sufficient conditions for guaranteeing the existence of a global attractor under small delays.
In this paper, we first prove that if a linear neutral functional differential equation is positive then it must degrade into a linear functional differential equation of retarded type. Then, we give some explicit criteria for positive linear functional differential equations. Consequently, we obtain a novel criterion for exponential stability of positive linear functional differential equations.
Basic results on semigroups on banach spaces.- Characterization of semigroups on banach spaces.- Spectral theory.- Asymptotics of semigroups on banach spaces.- Basic results on spaces Co(X).- Characterization of positive semigroups on Co(X).- Spectral theory of positive semigroups on Co(X).- Asymptotics of positive semigroups on Co(X).- Basic results on banach lattices and positive operators.- Characterization of positive semigroups on banach lattices.- Spectral theory of positive semigroups on banach lattices.- Asymptotics of positive semigroups on banach lattices.- Basic results on semigroups and operator algebras.- Characterization of positive semigroups on w*-algebras.- Spectral theory of positive semigroups on w*-algebras and their preduals.- Asymptotics of positive semigroups on c*-and w*-algebras.
We first introduce a class of positive linear Volterra difference equations. Then, we offer explicit criteria for uniform asymptotic stability of positive equations. Furthermore, we get a new Perron-Frobenius theorem for positive linear Volterra difference equations. Finally, we study robust stability of positive equations under structured perturbations and affine perturbations. Two explicit stability bounds with respect to these perturbations are given. Copyright (c) 2008 John Wiley & Sons, Ltd.
In this paper, we obtain exact non-exponential rates of growth and decay of solutions of linear functional differential equations
with unbounded delay. As a by-product, exponential asymptotic stability is ruled out for asymptotically stable solutions of
these equations. Equations with maximum functionals on the right hand side, as well as perturbed equations, are considered.
We also give examples showing how the rate of growth or decay of solutions depends on the rate of growth of the unbounded
delay. The results established here are obtained by constructing carefully functions which satisfy upper and lower differential
inequalities and one aim of the paper is to elucidate this constructive comparison technique.
KeywordsFunctional differential equation–Unbounded delay exponential asymptotic stability–Asymptotic stability–Halanay’s inequality–Equations with maxima–Pantograph equation
We first introduce the notion of positive linear Volterra-Stieltjes differential systems. Then, we give some characterizations
of positive systems. An explicit criterion and a Perron-Frobenius type theorem for positive linear Volterra-Stieltjes differential
systems are given. Next, we offer a new criterion for uniformly asymptotic stability of positive systems. Finally, we study
stability radii of positive linear Volterra-Stieltjes differential systems. It is proved that complex, real and positive stability
radius of positive linear Volterra-Stieltjes differential systems under structured perturbations coincide and can be computed
by an explicit formula. The obtained results in this paper include ones established recently for positive linear Volterra
integro-differential systems [36] and for positive linear functional differential systems [32]-[35] as particular cases. Moreover,
to the best of our knowledge, most of them are new.
This paper establishes the existence-and-uniqueness theorem of neutral stochastic functional differential equations with infinite delay and examines the almost sure stability of this solution with general decay rate. This result may be used to examine almost sure robust stability. To illustrate our idea more carefully, we carefully discuss a scalar stochastic integro-differential equation with neutral type and its asymptotic stability, including the exponential stability and the polynomial stability.
The paper attempts to solve a problem which was called a “challenge for the future” in Linear Algebra Appl. We define and investigate a new quantity for real matrices, the sign-real spectral radius, and derive various characterizations, bounds, and properties of it. In certain aspects our quantity shows similar behavior to the Perron root of a nonnegative matrix. It is shown that our quantity also has intimate connections to the componentwise distance to the nearest singular matrix. Relations to the Perron root of the (entrywise) absolute value of the matrix and to the μ-number are given as well.
Some asymptotic stability criteria are derived for systems of nonlinear functional differential equations with unbounded delays. The criteria are described as matrix equations or matrix inequalities, which are computationally flexible and efficient. The theories are then applied to the stabilization of time-delay systems via standard feedback control (SFC) or time-delayed feedback control (DFC). Several examples are given to illustrate the results.
We prove the existence of attractors for some types of differential problems containing infinite delays. Applications and examples are provided to illustrate the theory, which is valid for both cases with and without explicit dependence of time, and with or without uniqueness of solutions, as well.
In this work, we give an extension of the classical Perron–Frobenius theorem to positive quasi-polynomial matrices. Then the result obtained is applied to derive necessary and sufficient conditions for the exponential stability of positive linear time-delay differential systems.
We first introduce the notion of positive linear Volterra integral equations. Then, we offer a criterion for positive equations in terms of the resolvent. In particular, equations with nonnegative kernels are positive. Next, we obtain a variant of the Paley–Wiener theorem for equations of this class and its extension to perturbed equations. Furthermore, we get a Perron–Frobenius type theorem for linear Volterra integral equations with nonnegative kernels. Finally, we give a criterion for positivity of the initial function semigroup of linear Volterra integral equations and provide a necessary and sufficient condition for exponential stability of the semigroups.
An explicit criterion for positivity of the solution semigroup of linear Volterra integro-differential systems with infinitely many delays is given. Then exponential asymptotic stability is studied, and it is shown that, roughly speaking, a linear Volterra integro-differential system with delay is exponentially asymptotically stable if its characteristic equation has no zeros in the closed right half complex plane and the “kernel part” of the system exponentially decays. In particular, some simple criteria for exponential asymptotic stability of positive systems are presented. Finally, two sufficient conditions for exponential asymptotic stability of positive systems subjected to structured perturbations and affine perturbations are given. A simple example is given to illustrate the obtained results.
We first introduce the notion of positive linear Volterra integrodifferential equations. Then we give some characterizations of these positive equations. An explicit criterion and a Perron-Frobenius-type theorem for positive linear Volterra integrodifferential equations are given. Then we offer a new criterion for uniformly asymptotic stability of positive equations. Finally, we study stability radii of positive linear Volterra integrodifferential equations. It is proved that complex, real, and positive stability radii of positive linear Volterra equations under structured perturbations (or affine perturbations) coincide and can be computed by explicit formulae. To the best of our knowledge, most of the results of this paper are new.
Linear Volterra-Stieltjes differential equations are considered. We give a sufficient condition which ensures that the uniform asymptotic stability and the exponential asymptotic stability of linear Volterra-Stieltjes differential equations coincide. In particular, this yields necessary and sufficient conditions for the exponential asymptotic stability of linear Volterra-Stieltjes differential equations whose kernels are monotone. Then the class of positive linear Volterra-Stieltjes differential equations is studied in detail and simple criteria for the exponential asymptotic stability of positive equations are presented. Furthermore, some sufficient conditions for the exponential asymptotic stability of positive equations subjected to structured perturbations and affine perturbations are provided. Finally, by applying obtained results to linear Volterra integro-differential equations with infinite delay, we get explicit criteria for the exponential asymptotic stability of positive equations with delay. To the best of our knowledge, most of the obtained results of this paper are new.
We first give a sufficient condition for positivity of the solution semigroup of linear functional difference equations. Then,
we obtain a Perron–Frobenius theorem for positive linear functional difference equations. Next, we offer a new explicit criterion
for exponential stability of a wide class of positive equations. Finally, we study stability radii of positive linear functional
difference equations. It is proved that complex, real and positive stability radius of positive equations under structured
perturbations (or affine perturbations) coincide and can be computed by explicit formulae.
The material of this book is the outgrowth of a course of lectures which I have given from time to time at Harvard University on the subject of Dirichlet series and Laplace integrals. It is designed for students who have such a knowledge of analysis as might be obtained by reading the more fundamental parts of the familiar text of E. C. Titchmarsh on the theory of functions. I have taken pains to include proofs of results which such a student might not know even though they might be available elsewhere. For example, the first chapter is devoted largely to the study of the Riemann-Stieltjes integral. Although this material is in constant use by analysts it seems not to have been collected in convenient form. There are only a few instances where I have had to depart from this aim of having the book complete in itself. If he desires, the student may omit such parts without losing the fundamental ideas.
