Viable control for uncertain nonlinear dynamical systems described by differential inclusions

In this paper, we will study the viable control problem for a class of uncertain nonlinear dynamical systems described by a differential inclusion. The goal is to construct a feedback control such that all trajectories of the system are viable in a map. Moreover, for any initial states no viable in the map, under the feedback control, all solutions of the system are steered to the map with an exponential convergence rate and viable in the map after a finite time T. In this case, an estimate of the time T of all trajectories attaining the map is given. In the nanomedicine system, an example inspired from cerebral embolism and cerebral thrombosis problems illustrates the use of our main results.     

On the existence of periodic solutions for differential inclusions

In this paper we study the existence of periodic solutions for differential inclusions. We prove existence theorems under various sets of hypotheses for both the nonconvex and convex problems. Also we show the existence of extreme solutions. Some feedback control systems are also considered.     

Special symmetric periodic solutions of differential systems with distributed delay

We discuss the existence of periodic solutions to a system of differential equations with distributed delay which shows a certain type of symmetry. For this, such solutions are related to the solutions of a system of second-order ordinary differential equations.     

Periodic solutions for a class of differential inclusions in general Banach spaces

Let X be a real Banach space, an m-accretive operator and a multi-function which is 2π-periodic with respect to its first argument, has nonempty, closed, convex and weakly compact values and is strongly-weakly upper semicontinuous. In this paper we prove the existence of at least one solution for the problem

     

Computation of periodic solutions in maximal monotone dynamical systems with guaranteed consistency

In this paper, we study a class of set-valued dynamical systems that satisfy maximal monotonicity properties. This class includes linear relay systems, linear complementarity systems, and linear mechanical systems with dry friction under some conditions. We discuss two numerical schemes based on time-stepping methods for the computation of the periodic solutions when these systems are periodically excited. We provide formal mathematical justifications for the numerical schemes in the sense of consistency, which means that the continuous-time interpolations of the numerical solutions converge to the continuous-time periodic solution when the discretization step vanishes. The two time-stepping methods are applied for the computation of the periodic solution exhibited by a power electronic converter and the corresponding methods are compared in terms of approximation accuracy and computation time.     

Linear stability of general linear methods for systems of neutral delay differential equations

This paper is concerned with the numerical solution of delay differential equations (DDEs). We focus on the stability of general linear methods for systems of neutral DDEs with multiple delays. A type of interpolation procedure is considered for general linear methods. Linear stability properties of general linear methods with this interpolation procedure are investigated. Many extant results are unified.     

Lower bounds of the topological entropy for nonautonomous dynamical systems

In this paper, lower bounds of the topological entropy for nonautonomous dynamical systems are given via the growths of topological complexity in fundamental group and in degree.     

Existence and uniqueness of periodic solution for a class of differential systems

In this paper, sufficient conditions are obtained for the existence of a unique periodic solution for a class of differential systems.     

On the continuity of the Lyapunov functions in the converse stability theorems for discontinuous dynamical systems

In [H. Ye, A.N. Michel, L. Hou, Stability theory for hybrid dynamical systems, IEEE Transactions on Automatic Control 43 (4) (1998) 461–474] we established, among other results, a set of sufficient conditions for the uniform asymptotic stability of invariant sets for discontinuous dynamical systems (DDS) defined on metric space, and under some additional minor assumptions, we also established a set of necessary conditions (a converse theorem). This converse theorem involves Lyapunov functions which need not necessarily be continuous. In the present paper, we show that under some additional very mild assumptions, the Lyapunov functions for the converse theorem need actually be continuous.     

On monotone method for first and second order periodic boundary value problems and periodic solutions of functional differential equations

In this paper, we show that the method of monotone iterative technique is valid to obtain two monotone sequences that converge uniformly to extremal solutions of first and second order periodic boundary value problems and periodic solutions of functional differential equations. We obtain some new results relative to the lower solution α and upper solution β with α?β.     

Contraction of orbits in random dynamical systems on the circle

The paper deals with a theoretical justification of the effect, observed in computer experiments, of convergence of orbits (without tending to any particular point) in random dynamical systems on the circle. The corresponding theorem is proved under certain assumptions satisfied, in particular, in some C ¹-open domain in the space of random dynamical systems.It follows from this theorem that the corresponding skew product has two invariant measurable sections, naturally called an attractor and a repeller. Moreover, it turns out that convergence of orbits and the uniqueness of a stationary measure, phenomena that are mutually exclusive in the case of a single map, typically coexist in random dynamical systems.Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 36–54, 2004Original Russian Text Copyright © by V. A. Kleptsyn and M. B. NalskiiSupported in part by RFBR grants 02-01-00482 and 02-01-22002 and CRDF grant RM1-2358-MO-02.     

A kind of nonnegative matrices and its application on the stability of discrete dynamical systems

In this paper we introduce a new kind of nonnegative matrices which is called (sp) matrices. We show that the zero solutions of a class of linear discrete dynamical systems are asymptotically stable if and only if the coefficient matrices are (sp) matrices. To determine that a matrix is (sp) matrix or not is very simple, we need only to verify that some elements of the coefficient matrices are zero or not. According to the result above, we obtain the conditions for the stability of several classes of discrete dynamical systems.     

On the connectedness and asymptotic behaviour of solutions of reaction-diffusion systems

In this paper we consider reaction-diffusion systems in which the conditions imposed on the nonlinearity provide global existence of solutions of the Cauchy problem, but not uniqueness. We prove first that for the set of all weak solutions the Kneser property holds, that is, that the set of values attained by the solutions at every moment of time is compact and connected. Further, we prove the existence and connectedness of a global attractor in both the autonomous and nonautonomous cases. The obtained results are applied to several models of physical (or chemical) interest: a model of fractional-order chemical autocatalysis with decay, the Fitz-Hugh-Nagumo equation and the Ginzburg-Landau equation.     

Invariant manifolds, global attractors, almost automorphic and almost periodic solutions of non-autonomous differential equations

The paper is devoted to the study of non-autonomous evolution equations: invariant manifolds, compact global attractors, almost periodic and almost automorphic solutions. We study this problem in the framework of general non-autonomous (cocycle) dynamical systems. First, we prove that under some conditions such systems admit an invariant continuous section (an invariant manifold). Then, we obtain the conditions for the existence of a compact global attractor and characterize its structure. Third, we derive a criterion for the existence of almost periodic and almost automorphic solutions of different classes of non-autonomous differential equations (both ODEs (in finite and infinite spaces) and PDEs).     

On the oscillation of nonlinear two-dimensional differential systems

An oscillation criterion is given for a certain form of nonlinear two-dimensional differential systems. This criterion originated in a well-known oscillation result due to Coles (as extended and improved by Wong) concerning second order nonlinear differential equations with alternating coefficients.

     

Oscillations and multiple periodic solutions in systems of differential equations with piecewise continous delay

A system of two first-order liner differential equations with piecewise continuous delay is studied. The delay generates unusually interesting oscillation and periodic properties of the system. In particular nonlinear phenomena such as simultaneous existence of periodic solutions with different periods observed in linear delay systems     

On the existence and multiplicity of positive periodic solutions for first-order vector differential equation

In this paper, we consider the existence and multiplicity of positive periodic solutions for first-order vector differential equation x^′(t)+f(t,x(t))=0, a.e. t∈[0,ω] under the periodic boundary value condition x(0)=x(ω). Here ω is a positive constant, and is a Carathéodory function. Some existence and multiplicity results of positive periodic solutions are derived by using a fixed point theorem in cones.     

ON THE STABILITY OF DIFFERENTIAL SYSTEMS WITH TIME LAG

In this paer the inequality of Lemma 1 of [1] is extended.By means of our inequality and the method of Lyapunov function we study the stability of two kinds of large seate differential systems with time lag and the stability of a higher -order differential equation with time lag.The sufficient conditions for the stability(S.),the asymptotic stability(A.S.),the uniformly asymptotic stability(U.A.S) and the exponential asymptotic stability(E.A.S.) of the zero solutions of the systerms are obtained respectively.