Influence of anti-diagonals on the asymptotic stability for linear differential systems

Sufficient conditions are given for asymptotic stability of the linear differential system x′  =  B(t)x with B(t) being a 2  ×  2 matrix. All components of B(t) are not assumed to be positive. The matrix B(t) is naturally divisible into a diagonal matrix D(t) and an anti-diagonal matrix A(t). Our concern is to clarify a positive effect of the anti-diagonal part A(t)x on the asymptotic stability for the system x′  =  B(t)x.      

On a criterion for the complete controllability of nonautonomous linear systems

We describe the controllability sets of linear nonautonomous systems ẋ = A(t)x + B(t)u, x ∈ ℝ ⁿ , u ∈ U ⊆ ℝ ^m , with entire matrix functions A(t) and B(t) and with a linear set U of control constraints. We derive a criterion for the complete controllability of these linear systems in terms of derivatives of the entire matrix functions A(t) and B(t) at zero. This complete controllability criterion is compared with the Kalman and Krasovskii criteria.     

On the Instability of Linear Nonautonomous Delay Systems

The unstable properties of the linear nonautonomous delay system x(t) = A(t)x(t) + B(t)x(t – r(t)), with nonconstant delay r(t), are studied. It is assumed that the linear system y(t) = (A(t) + B(t))y(t) is unstable, the instability being characterized by a nonstable manifold defined from a dichotomy to this linear system. The delay r(t) is assumed to be continuous and bounded. Two kinds of results are given, those concerning conditions that do not include the properties of the delay function r(t) and the results depending on the asymptotic properties of the delay function.     

Stability Theory of General Contractions for Delay Equations

We study the persistence of the asymptotic stability of delay equations both under linear and nonlinear perturbations. Namely, we consider nonautonomous linear delay equations v′ = L(t)v _t with a nonuniform exponential contraction. Our main objective is to establish the persistence of the nonuniform exponential stability of the zero solution both under nonautonomous linear perturbations, i.e., for the equation v′ = (L(t) + M(t))v _t , thus discussing the so-called robustness problem, and under a large class of nonlinear perturbations, namely for the equation v′ = L(t)v _t + f(t, v _t ). In addition, we consider general contractions e ^−λρ(t) determined by an increasing function ρ that includes the usual exponential behavior with ρ(t) = t as a very special case. We also obtain corresponding results in the case of discrete time.     

线性时滞微分方程稳定性分析和数值例子

§ 1　IntroductionFunctional differential equations have a wide range of applications in science andengineering.The simplestand perhapsmostnatural type of functional differential equationis a“delay differential equation”,that is,differential equation with dependence on the paststate.The simplest type of pastdependence is thatit is carried through the state variablebut not through its derivative.Then the equation can be expressed as delay differentialequations(DDEs) .There are also a number…     

On the asymptotic stability for impulsive functional differential equations

We consider the asymptotic stability problems by Lyapunov functionals V for a class of functional differential equations with impulses of the form x′(t)=f(t,x _t ), x∈R ⁿ , t≧t ₀, t≠t _k ; △x=I _k (t,x(t ⁻)), t=t _k , k∈Z ⁺ . Some new asymptotic stability results are presented by using an idea originated by Burton and Makay [6] and developed by Zhang [1]. We generalize some known results about impulsive functional differential equations in the literature in which we only require the derivative of V to be negative definite on a sequence of intervals I _n =[s _n ,ξ _n ] which may or may not be contained in the sequence of impulsive time intervals [t _n ,t _n+1).     

Asymptotic preconditioning of linear homogeneous systems of differential equations

We consider the asymptotic behavior of solutions of a linear differential system x^′=A(t)x, where A is continuous on an interval ([a,∞). We are interested in the situation where the system may not have a desirable asymptotic property such as stability, strict stability, uniform stability, or linear asymptotic equilibrium, but its solutions can be written as x=Pu, where P is continuously differentiable on [a,∞) and u is a solution of a system u^′=B(t)u that has the property in question. In this case we say that P preconditions the given system for the property in question.     

On perturbations of differentiable semigroups

LetX be a Banach space and letA be the infinitesimal generator of a differentiable semigroup {T(t) |t ≥ 0}, i.e. aC ₀-semigroup such thatt ↦T(t)x is differentiable on (0, ∞) for everyx εX. LetB be a bounded linear operator onX and let {S(t) |t ≥ 0} be the semigroup generated byA +B. Renardy recently gave an example which shows that {S(t) |t ≥ 0} need not be differentiable. In this paper we give a condition on the growth of ‖T′(t)‖ ast ↓ 0 which is sufficient to ensure that {S(t) |t ≥ 0} is differentiable. Moreover, we use Renardy’s example to study the optimality of our growth condition. Our results can be summarized roughly as follows:

Global stabilization of nonlinear systems by quadratic Lyapunov functions

The system x = A (t, x)x + B(t, x)u, where A(t, x) and B(t, x) are, respectively, n × n and n × m (m<n) continuous matrices whose elements are uniformly bounded for t ≽ t ₀ and x ∈ ℝ ⁿ , is considered. It is assumed that the system has relative degree q = n - m + 1, and the determinant of the matrix composed of the last m rows of the matrix B(t, x) is bounded away from zero for t ≽ t ₀ and x ∈ ℝ ⁿ . A special quadratic Lyapunov function with constant positive definite coefficient matrix H depending only on the range of variation of the coefficients in the matrices A(t, x) and B(t, x) is constructed and applied to obtain a control u(t, x) =7n ~B⋆ (t, x)H depending on a scalar parameter 7n under which the system is globally asymptotically stable provided that it is closed. Here, ~B (t, x) is the scalar matrix obtained from the matrix B(t, x) by setting the first n - m rows to zero.     

Asymptotic stability and boundedness for functional differential equations

We study a system(D)x′=F(t,x _t) of functional differential equations, together with a scalar equation(S)x′=−a(t)f(x)+b(t)g(x(t−h)) as well as perturbed forms. A Liapunov functional is constructed which has a derivative of a nature that has been widely discussed in the literature. On the basis of this example we prove results for (D) on asymptotic stability and equi-boundedness. Supported in part by NSF of China, Key Project # 19331060     

A sum formula for the Casson-Walker invariant

We study the following question: How does the Casson-Walker invariant λ of a rational homology 3-sphere obtained by gluing two pieces along a surface depend on the two pieces? Our partial answer may be stated as follows. For a compact oriented 3-manifold A with boundary ∂A, the kernel L _A of the map from H ₁(∂A;Q) to H ₁(A;Q) induced by the inclusion is called the Lagrangian of A. Let Σ be a closed oriented surface, and let A, A′, B and B′ be four rational homology handlebodies such that ∂A, ∂A′, −∂B and −∂B′ are identified via orientation-preserving homeomorphisms with Σ. Assume that L _A = L _{A ′} and L _B = L _{B ′} inside H ₁(Σ;Q) and also assume that L _A and L _B are transverse. Then we express

EXISTENCE AND UNIQUENESS FOR SECOND-ORDER VECTOR BOUNDARY VALUE PROBLEM OF NONLINEAR SYSTEMS

This paper is concerned with the following second-order vector boundary value problem ：x^R=f（t,Sx,x,x＇）,0〈t〈1,x（0）=A,g（x（1）,x＇（1））=B,where x,f,g,A and B are n-vectors. Under appropriate assumptions,existence and uniqueness of solutions are obtained by using upper and lower solutions method.     

Non-uniform stability for bounded semi-groups on Banach spaces

Let S(t) be a bounded strongly continuous semi-group on a Banach space B and – A be its generator. We say that S(t) is semi-uniformly stable when S(t)(A + 1)⁻¹ tends to 0 in operator norm. This notion of asymptotic stability is stronger than pointwise stability, but strictly weaker than uniform stability, and generalizes the known logarithmic, polynomial and exponential stabilities. In this note we show that if S is semi-uniformly stable then the spectrum of A does not intersect the imaginary axis. The converse is already known, but we give an estimate on the rate of decay of S(t)(A + 1)⁻¹, linking the decay to the behaviour of the resolvent of A on the imaginary axis. This generalizes results of Lebeau and Burq (in the case of logarithmic stability) and Liu-Rao and Bátkai-Engel-Prüss-Schnaubelt (in the case of polynomial stability). This work was partially supported by the French ANR ControlFlux. The second author would like to thank Nicolas Burq for fruitful discussions on the subject, and Luc Miller for pointing out the stability theorem of Lyubich, Vũ, Arendt and Batty and the article [3].     

Canonical form with respect to semiscalar equivalence for a matrix pencil with nonsingular first matrix

Polynomial n × n matrices A(x) and B(x) over a field \mathbbF \mathbb{F} are called semiscalar equivalent if there exist a nonsingular n × n matrix P over \mathbbF \mathbb{F} and an invertible n × n matrix Q(x) over \mathbbF \mathbb{F} [x] such that A(x) = PB(x)Q(x). We give a canonical form with respect to semiscalar equivalence for a matrix pencil A(x) = A _0x - A ₁, where A ₀ and A ₁ are n × n matrices over \mathbbF \mathbb{F} , and A ₀ is nonsingular.     

Controlling the space with preassigned responses

It is shown that large classes of control systems, which include certain systems of the typex+A(t)x=B(t)u, can be handled in such a way that the control functionsu(t) are actually associated with responsesx(t) that belong to known families of functions. In particular, it is possible, for a variety of perturbationsB(t)u and operatorsA(t) with convex domains, to have responses that are line segments joining the origin to the reachable states.The present approach establishes the fact that a vast number of results from functional analysis concerning ranges of operators can be effectively applied to the general theory of control. It is also rather significant that the present theory does not necessarily require the solvability of the associated Cauchy problem.The operatorsB(t)u do not have to be invertible inu. However, it is shown that continuous controlsu(t) can be obtained for a variety of problems whenB ^–1(t)u exists and is continuous int.     

Diffusion approximation for the convection-diffusion equation with random drift

We consider the asymptotic behavior of the solutions ofscaled convection-diffusion equations ∂ _t u _ɛ (t, x) = κΔ _x (t, x) + 1/ɛV(t/ɛ²,xɛ) ·∇ _x u _ɛ (t, x) with the initial condition u _ɛ(0,x) = u ₀(x) as the parameter ɛ↓ 0. Under the assumptions that κ > 0 and V(t, x), (t, x) ∈R ^d is a d-dimensional,stationary, zero mean, incompressible, Gaussian random field, Markovian and mixing in t we show that the laws of u _ɛ(t,·), t≥ 0 in an appropriate functional space converge weakly, as ɛ↓ 0, to a δ-type measureconcentrated on a solution of a certain constant coefficient heat equation. Received: 23 March 2000 / Revised version: 5 March 2001 / Published online: 9 October 2001     

Stability of <Emphasis Type="Italic">C</Emphasis>-regularized Semigroups

Let T = (T(t))t≥0 be a bounded C-regularized semigroup generated by A on a Banach space X and R(C) be dense in X. We show that if there is a dense subspace Y of X such that for every x ∈ Y, σu(A, Cx), the set of all points λ ∈ iR to which (λ - A)^-1 Cx can not be extended holomorphically, is at most countable and σr(A) N iR = Ф, then T is stable. A stability result for the case of R(C) being non-dense is also given. Our results generalize the work on the stability of strongly continuous senfigroups.     

Approximate weak invariance for semilinear differential inclusions in Banach spaces

In this paper we give a criterion for a given set K in Banach space to be approximately weakly invariant with respect to the differential inclusion x′(t) ∈ Ax(t) + F(x(t)), where A generates a C ₀-semigroup and F is a given multi-function, using the concept of a tangent set to another set. As an application, we establish the relation between approximate solutions to the considered differential inclusion and solutions to the relaxed one, i.e., x′(t) ∈ Ax(t) + [`(co)]\overline {co} F(x(t)), without any Lipschitz conditions on the multi-function F.     

Oscillation criteria for nonlinear second-order differential equations with damping

Some new oscillation criteria are given for general nonlinear second-order ordinary differential equations with damping of the form x″+ p ( t ) x′+ q ( t ) f ( x ) = 0, where f is monotone or nonmonotone. Our results generalize and extend some earlier results of Deng. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 5, pp. 694–700, May, 2008.     

Probabilities of high excursions of Gaussian fields

Let {ξ(t), t ∈ T} be a differentiable (in the mean-square sense) Gaussian random field with E ξ(t) ≡ 0, D ξ(t) ≡ 1, and continuous trajectories defined on the m-dimensional interval T ì \mathbbR^m T \subset {\mathbb{R}^m} . The paper is devoted to the problem of large excursions of the random field ξ. In particular, the asymptotic properties of the probability P = P{−v(t) < ξ(t) < u(t), t ∈ T}, when, for all t ∈ T, u(t), v(t) ⩾ χ, χ → ∞, are investigated. The work is a continuation of Rudzkis research started in [R. Rudzkis, Probabilities of large excursions of empirical processes and fields, Sov. Math., Dokl., 45(1):226–228, 1992]. It is shown that if the random field ξ satisfies certain smoothness and regularity conditions, then P = e^−Q  + Qo(1), where Q is a certain constructive functional depending on u, v, T, and the matrix function R(t) = cov(ξ′(t), ξ′(t)).