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1.
Using the white noise setting, in particular the Wick product, the Hermite transform, and the Kondratiev space, we present
a new approach to study linear stochastic systems, where randomness is also included in the transfer function. We prove BIBO
type stability theorems for these systems, both in the discrete and continuous time cases. We also consider the case of dissipative
systems for both discrete and continuous time systems.We further study ℓ
1-ℓ
2 stability in the discrete time case, and L
2-L
∞ stability in the continuous time case. 相似文献
2.
F. D. Pryashnikov A. I. Grushun T. A. Grushun 《Journal of Mathematical Sciences》1996,82(3):3409-3411
We introduce the concept of a representative set as the set of values of the stability and quality characteristics possessing
the properties of informativeness and nonredundancy. We obtain relations that determine representative sets for problems of
constructing domains of stability and quality for linear stationary systems and retarded systems.
Translated fromDinamicheskie Sistemy, No. 13, 1994, pp. 16–20. 相似文献
3.
D. H. Korenivs’kyi 《Ukrainian Mathematical Journal》2005,57(12):2021-2026
We describe the destabilizing (in the sense of a decrease in the reserve of mean-square asymptotic stability) effect of random
parametric perturbations of the white-noise type in quasilinear continuous and discrete dynamical systems (Lur’e-Postnikov
systems of automatic control with nonlinear feedback). We use stochastic Lyapunov functions in the form of linear combinations
of the types “a quadratic form of phase coordinates plus the integral of a nonlinearity” (continuous systems) and “a quadratic
form of phase coordinates plus the integral sum for a nonlinearity” (discrete systems) and the matrix algebraic Sylvester
equations associated with stochastic Lyapunov functions of this form.
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 12, pp. 1719–1724, December, 2005. 相似文献
4.
V. V. Fomichev 《Computational Mathematics and Modeling》1998,9(3):260-267
We study the problem of inversion of linear finite-dimensional dynamic systems in terms of the output. For different types
of systems we propose several algorithms for inversion. We study the stability of the algorithms with respect to various perturbations.
Translated fromAlgoritmy Upravleniya i Identifikatsii, pp. 156–167, 1997. 相似文献
5.
E. A. Barabanov 《Differential Equations》2010,46(6):798-807
We consider families of linear differential systems depending on a real parameter that occurs only as a factor multiplying
the matrix of the system. The asymptotic stability set of such a family is defined as the set of all parameter values for
which the corresponding systems in the family are asymptotically stable. We prove that a set on the real axis is the asymptotic
stability set of such a family if and only if it is an F
σδ
-set lying entirely on an open ray with origin at zero. In addition, for any set of this kind, the coefficient matrix of a
family whose asymptotic stability set coincides with this set can be chosen to be infinitely differentiable and uniformly
bounded on the time half-line. 相似文献
6.
We consider a system of linear ordinary differential equations in which the coefficient matrix multiplying the derivative
of the unknown vector function is identically singular. For systems with constant and variable coefficients, we obtain nonresonance
criteria (criteria for bounded-input bounded-output stability). For single-input control systems, we consider the problem
of synthesizing a nonresonant system in the stationary and nonstationary cases. An arbitrarily high unsolvability index is
admitted. The analysis is carried out under assumptions providing the existence of a so-called “equivalent form” with separated
“algebraic” and “differential” components. 相似文献
7.
E. A. Barabanov 《Differential Equations》2010,46(5):613-627
We consider families of linear differential systems continuously depending on a real parameter. The stability (respectively,
asymptotic stability) set of such a family is defined as the set of all values of the parameter for which the corresponding
systems in the family are stable (respectively, asymptotically stable). We show that a set on the real axis is the stability
(respectively, asymptotic stability) set of some family of this kind if and only if it is an F
σ
-set (respectively, an F
σδ
-set). For families in which the parameter occurs only as a factor multiplying the matrix of the system, their stability sets
are exactly F
σ
-sets containing zero on the real line. The asymptotic stability sets of such families will be described in the second part
of the present paper. 相似文献
8.
We study finite difference discretizations of initial boundary value problems for linear symmetric hyperbolic systems of equations
in multiple space dimensions. The goal is to prove stability for SBP-SAT (Summation by Parts—Simultaneous Approximation Term)
finite difference schemes for equations with variable coefficients. We show stability by providing a proof for the principle
of frozen coefficients, i.e., showing that variable coefficient discretization is stable provided that all corresponding constant
coefficient discretizations are stable. 相似文献
9.
This paper studies the stability of a three‐station fluid network. We show that, unlike the two‐station networks in Dai and
Vande Vate [18], the global stability region of our three‐station network is not the intersection of its stability regions
under static buffer priority disciplines. Thus, the “worst” or extremal disciplines are not static buffer priority disciplines.
We also prove that the global stability region of our three‐station network is not monotone in the service times and so, we
may move a service time vector out of the global stability region by reducing the service time for a class. We introduce the
monotone global stability region and show that a linear program (LP) related to a piecewise linear Lyapunov function characterizes
this largest monotone subset of the global stability region for our three‐station network. We also show that the LP proposed
by Bertsimas et al. [1] does not characterize either the global stability region or even the monotone global stability region
of our three‐station network. Further, we demonstrate that the LP related to the linear Lyapunov function proposed by Chen
and Zhang [11] does not characterize the stability region of our three‐station network under a static buffer priority discipline.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
10.
D. G. Korenevskii 《Ukrainian Mathematical Journal》1998,50(8):1224-1232
We obtain spectral and algebraic coefficient criteria and sufficient conditions for the mean-square asymptotic stability of
solutions of systems of linear stochastic difference equations with continuous time and delay. We consider the case of a rational
correlation between delays and a “white-noise”-type stochastic perturbation of coefficients. We use the method of Lyapunov
functions. Most results are presented in terms of the Sylvester and Lyapunov matrix algebraic equations.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1073–1081, August, 1998.
This work was partially supported by the Joint Foundation of the Ukrainian Government and the Soros International Science
Foundation (grant No. K42100). 相似文献
11.
12.
Discretization of autonomous ordinary differential equationsby numerical methods might, for certain step sizes, generatesolution sequences not corresponding to the underlying flowso-calledspurious solutions or ghost solutions.In this paper we explain this phenomenon for the case of explicitRunge-Kutta methods by application of bifurcation theory fordiscrete dynamical systems. An important tool in our analysisis the domain of absolute stability, resulting from the applicationof the method to a linear test problem. We show that hyperbolicfixed points of the (nonlinear) differential equation are inheritedby the difference scheme induced by the numerical method whilethe stability type of these inherited genuine fixed points iscompletely determined by the method's domain of absolute stability.We prove that, for small step sizes, the inherited fixed pointsexhibit the correct stability type, and we compute the correspondinglimit step size. Moreover, we show in which way the bifurcationsoccurring at the limit step size are connected to the valuesof the stability function on the boundary of the domain of absolutestability, where we pay special attention to bifurcations leadingto spurious solutions. In order to explain a certain kind ofspurious fixed points which are not connected to the set ofgenuine fixed points, we interprete the domain of absolute stabilityas a Mandeibrot set and generalize this approach to nonlinearproblems. 相似文献
13.
We consider implicit integration methods for the solution of stiff initial value problems for second-order differential equations
of the special form y' = f(y). In implicit methods, we are faced with the problem of solving systems of implicit relations. This paper focuses on the construction
and analysis of iterative solution methods which are effective in cases where the Jacobian of the right‐hand side of the differential
equation can be split into a sum of matrices with a simple structure. These iterative methods consist of the modified Newton
method and an iterative linear solver to deal with the linear Newton systems. The linear solver is based on the approximate
factorization of the system matrix associated with the linear Newton systems. A number of convergence results are derived
for the linear solver in the case where the Jacobian matrix can be split into commuting matrices. Such problems often arise
in the spatial discretization of time‐dependent partial differential equations. Furthermore, the stability matrix and the
order of accuracy of the integration process are derived in the case of a finite number of iterations.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
14.
A. M. Samoilenko 《Ukrainian Mathematical Journal》2011,63(2):278-314
We consider problems of the linear theory of systems of ordinary differential equations related to the investigation of invariant
hyperplanes of these systems, the notion of equivalence for these systems, and the Floquet–Lyapunov theory for periodic systems
of linear equations. In particular, we introduce the notion of equivalence of systems of linear differential equations of
different orders, propose a new formula of the Floquet form for periodic systems, and present the application of this formula
to the introduction of amplitude–phase coordinates in a neighborhood of a periodic trajectory of a dynamical system. 相似文献
15.
In the present paper, we consider linear stationary dynamical systems over a Boolean semiring B. We analyze the complete observability,
identifiability, reachability, and controllability of such systems. We define the notion of a “graph of modules” of completely
controllable, completely reachable Boolean linear stationary systems by analogy with the spaces of modules in the case of
systems over fields. We give a graph-theoretic interpretation of systems of this class. We solve the isomorphism problem in
this class of systems. 相似文献
16.
Cícero F. Carvalho 《manuscripta mathematica》1999,98(2):155-163
We start this work by studying free linear systems on singular curves and related base point free linear systems on the non-singular
model. We apply these results to the study of pencils of small degree on non-singular curves. We also prove a “base point
free pencil trick” which holds for any (possibly) singular curve.
Received: 15 June 1998 相似文献
17.
We deal with linear multi-step methods for SDEs and study when the numerical approximation shares asymptotic properties in
the mean-square sense of the exact solution. As in deterministic numerical analysis we use a linear time-invariant test equation
and perform a linear stability analysis. Standard approaches used either to analyse deterministic multi-step methods or stochastic
one-step methods do not carry over to stochastic multi-step schemes. In order to obtain sufficient conditions for asymptotic
mean-square stability of stochastic linear two-step-Maruyama methods we construct and apply Lyapunov-type functionals. In
particular we study the asymptotic mean-square stability of stochastic counterparts of two-step Adams–Bashforth- and Adams–Moulton-methods,
the Milne–Simpson method and the BDF method.
AMS subject classification (2000) 60H35, 65C30, 65L06, 65L20 相似文献
18.
19.
Adina Luminita Sasu Bogdan Sasu 《Proceedings of the American Mathematical Society》2004,132(12):3653-3659
We introduce and characterize the stability radius of systems whose state evolution is described by linear skew-product semiflows. We obtain a lower bound for the stability radius in terms of the Perron operators associated to the linear skew-product semiflow. We generalize a result due to Hinrichsen and Pritchard.
20.
This paper addresses the robust stabilization and H
∞ control problem for a class of linear polytopic systems with continuously distributed delays. The control objective is to
design a robust H
∞ controller that satisfies some exponential stability constraints on the closed-loop poles. Using improved parameter-dependent
Lyapunov Krasovskii functionals, new delay-dependent conditions for the robust H
∞ control are established in terms of linear matrix inequalities. 相似文献