Stability radii of positive linear Volterra-Stieltjes equations

We study stability radii of linear Volterra-Stieltjes equations under multi-perturbations and affine perturbations. A lower and 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 functions. Then, the class of positive linear Volterra-Stieltjes equations 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. As direct consequences of the obtained results, we get some results on robust stability of positive linear integro-differential equations and of positive linear functional differential equations. To the best of our knowledge, most of the results of this paper are new.     

Positivity and stability of linear functional differential equations with infinite delay

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.     

Stability radius of linear parameter-varying systems and applications

In this paper, we present a unifying approach to the problems of computing of stability radii of positive linear systems. First, we study stability radii of linear time-invariant parameter-varying differential systems. A formula for the complex stability radius under multi perturbations is given. Then, under hypotheses of positivity of the system matrices, we prove that the complex, real and positive stability radii of the system under multi perturbations (or affine perturbations) coincide and they are computed via simple formulae. As applications, we consider problems of computing of (strong) stability radii of linear time-invariant time-delay differential systems and computing of stability radii of positive linear functional differential equations under multi perturbations and affine perturbations. We show that for a class of positive linear time-delay differential systems, the stability radii of the system under multi perturbations (or affine perturbations) are equal to the strong stability radii. Next, we prove that the stability radii of a positive linear functional differential equation under multi perturbations (or affine perturbations) are equal to those of the associated linear time-invariant parameter-varying differential system. In particular, we get back some explicit formulas for these stability radii which are given recently in [P.H.A. Ngoc, Strong stability radii of positive linear time-delay systems, Internat. J. Robust Nonlinear Control 15 (2005) 459-472; P.H.A. Ngoc, N.K. Son, Stability radii of positive linear functional differential equations under multi perturbations, SIAM J. Control Optim. 43 (2005) 2278-2295]. Finally, we give two examples to illustrate the obtained results.     

On stability and robust stability of positive linear Volterra equations in Banach lattices

We study positive linear Volterra integro-differential equations in Banach lattices. A characterization of positive equations is given. Furthermore, an explicit spectral criterion for uniformly asymptotic stability of positive equations is presented. Finally, we deal with problems of robust stability of positive systems under structured perturbations. Some explicit stability bounds with respect to these perturbations are given.     

Stability in first approximation of stochastic systems with after effect : PMM vol. 40, n 6, 1976, pp. 1116–1121

The theorem on existence of the Liapunov functionals and the theorem on stability in first approximation for a stochastic differential equation with aftereffect are proved.The suggestion of the replacement of Liapunov functions by functionals [1] in the investigation of the stability of ordinary differential equations with lag, has been widely utilized in dealing with determinate systems, as well as in the case of linear and nonlinear stochastic systems (see e. g. [2 – 11]). Results concerning the stability in the first approximation were obtained for stochastic systems in [12 – 18] and others. Use of Liapunov functionals for the differential equations with aftereffect was first encountered in [1, 19, 20] where the inversion theorems were proved and conditions for the stability in first approximation were obtained.Below a stochastic differential equation with aftereffect is investigated where the random perturbations represent an arbitrary process with independent increments.     

Order and stability properties of explicit multivalue methods

Recent papers by Burrage and Moss [1] and Burrage [2] have studied in some detail the order properties of implicit multivalue (or general linear) methods and certain classes of these methods were proposed as being suitable for solving stiff differential equations. In this present paper we study the order and stability of explicit multivalue methods with a view to deriving new families of methods suitablefor solving non stiff problems.     

Mean-square stability of second-order Runge–Kutta methods for multi-dimensional linear stochastic differential systems

In this paper, the mean-square stability of second-order Runge–Kutta schemes for multi-dimensional linear stochastic differential systems is studied. Motivated by the work of Tocino [Mean-square stability of second-order Runge–Kutta methods for stochastic differential equations, J. Comput. Appl. Math. 175 (2005) 355–367] and Saito and Mitsui [Mean-square stability of numerical schemes for stochastic differential systems, in: International Conference on SCIentific Computation and Differential Equations, July 29–August 3 2001, Vancouver, British Columbia, Canada] we investigate the mean-square stability of second-order Runge–Kutta schemes for multi-dimensional linear stochastic differential systems with one multiplicative noise. Stability criteria are established and numerical examples that confirm the theoretical results are also presented.     

Nonlinear functionals and a theorem of Sun

We use a recently developed theory of nonlinear functionals in the study of oscillations of second-order symmetric vector differential systems to extend a number of theorems of Sun [New Kamenev type theorems for second order linear matrix differential systems, Appl. Math. Lett., 2004, in press] under a common theme. The criteria presented here are of the form: the integral of the coefficient matrix is bounded at infinity (in a sense to be made explicit in the paper) and bounded away from a positive absolute constant implies oscillation at infinity.     

Characterizations of Positive Linear Volterra Integro-differential Systems

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,+∞). The first and last author are supported by the Japan Society for Promotion of Science (JSPS) ID No. P 05049.     

An Improved Theta-method for Systems of Ordinary Differential Equations

The / -method of order 1 or 2 (if / =1/2) is often used for the numerical solution of systems of ordinary differential equations. In the particular case of linear constant coefficient stiff systems the constraint 1/2 h / h 1, which excludes the explicit forward Euler method, is essential for the method to be A -stable. Moreover, unless / =1/2, this method is not elementary stable in the sense that its fixed-points do not display the linear stability properties of the fixed-points of the involved differential equation. We design a non-standard version of the / -method of the same order. We prove a result on the elementary stability of the new method, irrespective of the value of the parameter / ] [0,1]. Some absolute elementary stability properties pertinent to stiffness are discussed.     

Numerical detection of instability regions for delay models with delay-dependent parameters

In this paper we are interested in gaining local stability insights about the interior equilibria of delay models arising in biomathematics. The models share the property that the corresponding characteristic equations involve delay-dependent coefficients. The presence of such dependence requires the use of suitable criteria which usually makes the analytical work harder so that numerical techniques must be used. Most existing methods for studying stability switching of equilibria fail when applied to such a class of delay models. To this aim, an efficient criterion for stability switches was recently introduced in [E. Beretta, Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal. 33 (2002) 1144–1165] and extended [E. Beretta, Y. Tang, Extension of a geometric stability switch criterion, Funkcial Ekvac 46(3) (2003) 337–361]. We describe how to numerically detect the instability regions of positive equilibria by using such a criterion, considering both discrete and distributed delay models.     

Robust Stability of Positive Linear Systems Under Time-Varying Perturbations

By a novel approach, we get explicit robust stability bounds for positive linear differential systems subject to time-varying multi-perturbations and time-varying affine perturbations. Our approach is based on the celebrated Perron-Frobenius theorem and ideas of the comparison principle. An example is given to illustrate the obtained results.     

Polynomial stability of positive switched homogeneous systems with time delay

Based on the logarithm contraction average dwell-time method, this paper investigates the polynomial stability of positive switched homogeneous time-delay systems whose vector fields are of different degrees with respect to a dilation map. Using the analytical skills developed in positive systems, an explicit polynomial stability criterion is established for the first time for the involved system under the logarithm contraction average dwell-time switching. Moreover, the main result is applied to the polynomial stability of Persidskii-type switched systems.     

Global exponential stability of certain switched systems with time-varying delays

This paper is focused on global exponential stability of certain switched systems with time-varying delays. By using an average dwell time (ADT) approach that is different from the method in [P.H.A. Ngoc, On exponential stability of nonlinear differential systems with time-varying delay, Applied Mathematics Letters 25 (2012) 1208–1213], we establish a new global exponential stability criterion for the switched linear time-delay system under the ADT switching. We also apply this method to a general switched nonlinear time-delay system. A numerical example is given to show the effectiveness of our results.     

An improved stability criterion for a class of neutral differential equations

This work gives an improved criterion for asymptotical stability of a class of neutral differential equations. By introducing a new Lyapunov functional, we avoid the use of the stability assumption on the main operators and derive a novel stability criterion given in terms of a LMI, which is less restricted than that given by Park [J.H. Park, Delay-dependent criterion for asymptotic stability of a class of neutral equations, Appl. Math. Lett. 17 (2004) 1203–1206] and Sun et al. [Y.G. Sun, L. Wang, Note on asymptotic stability of a class of neutral differential equations, Appl. Math. Lett. 19 (2006) 949–953].     

Positive periodic solutions of competitive kolmogorov diffusion systems with interference constants

In this paper, we consider the positive periodic solutions of multispecies patches connected by discrete diffusion with a within-patch dynamics of periodic Kolmogorov type. A new criterion for existence and globally asymptotic stability of positive periodic solution is established. The results in [1–4] are improved and extended.     

Local first integrals of differential systems and diffeomorphisms

In this paper using theory of linear operators and normal forms we generalize a result of Poincaré [11] about the non-existence of local first integrals for systems of differential equations in a neighbourhood of a singular point. As an application of the generalized result, and under more weak conditions we obtain a result of Furta [8] about local first integrals of semi-quasi-homogeneous systems. Moreover, for diffeomorphisms and periodic differential systems we give definitions of their first integrals, and generalize the previous results about systems of differential equations to diffeomorphisms in a neighbourhood of a fixed point and to periodic differential systems in a neighbourhood of a constant solution.     

An explicit criterion for finite-time stability of linear nonautonomous systems with delays

In this paper, the problem of finite-time stability of linear nonautonomous systems with time-varying delays is considered. Using a novel approach based on some techniques developed for linear positive systems, we derive new explicit conditions in terms of matrix inequalities ensuring that the state trajectories of the system do not exceed a certain threshold over a pre-specified finite time interval. These conditions are shown to be relaxed for the Lyapunov asymptotic stability. A numerical example is given to illustrate the effectiveness of the obtained result.     

Expanding stability regions of explicit advective-diffusive finite difference methods by Jacobi preconditioning

Explicit time differencing methods for solving differential equations are advantageous in that they are easy to implement on a computer and are intrinsically very parallel. The disadvantage of explicit methods is the severe restrictions that are placed on stable time-step intervals. Stability bounds for explicit time differencing methods on advective–diffusive problems are generally determined by the diffusive part of the problem. These bounds are very small and implicit methods are used instead. The linear systems arising from these implicit methods are generally solved by iterative methods. In this article we develop a methodology for increasing the stability bounds of standard explicit finite differencing methods by combining explicit methods, implicit methods, and iterative methods in a novel way to generate new time-difference schemes, called preconditioned time-difference methods. A Jacobi preconditioned time differencing method is defined and analyzed for both diffusion and advection–diffusion equations. Several computational examples of both linear and nonlinear advective-diffusive problems are solved to demonstrate the accuracy and improved stability limits. © 1995 John Wiley & Sons, Inc.     

On the stability of rational Runge Kutta methods

This paper is concerned with the stability of rational two-stage Runge Kutta methods for the numerical solution of stiff differential systems. With a stability analysis based on linear diagonal systems of arbitrary dimension, we find necessary and sufficient conditions for the coefficients of a method to be A(α) and A(α, β) stable, extending previous results on this subject given by Hairer [1] and Wambecq [3], [4], [5].