Mean square stability analysis of impulsive stochastic differential equations with delays

In this article, based on Lyapunov–Krasovskii functional method and stochastic analysis theory, we obtain some new criteria ensuring mean square stability of trivial solution of a class of impulsive stochastic differential equations with delays. As an application, a class of stochastic impulsive neural network with delays has been discussed. One illustrative example has been provided to show the effectiveness of our results.     

Stabilization of linear control systems in Hilbert space with nonlinear feedback

This paper considers the problem of stabilization of linear control systems in Hilbert space with nonlinear feedback. Extensions of known results in the linear feedback case to the nonlinear one are given.     

Generalized joint spectral radius and stability of switching systems

This paper extends the notion of generalized joint spectral radius with exponents, originally defined for a finite set of matrices, to probability distributions. We show that, under a certain invariance condition, the radius is calculated as the spectral radius of a matrix that can be easily computed, extending the classical counterpart. Using this result we investigate the mean stability of switching systems. In particular we establish the equivalence of mean square stability, simultaneous contractibility in square mean, and the existence of a quadratic Lyapunov function. Also the stabilization of positive switching systems is studied. Numerical examples are given to illustrate the results.     

Multi-synchronization of chaos via linear output feedback strategy

Multi-synchronization of chaotic systems based on the master-slave scheme as an extension of the dual synchronization problem is introduced. It is assumed that the only information available from the master systems is a linear combination of their state vectors. The design procedure for multi-synchronization through output feedback strategy is described and the sufficient condition is given. The performance of the proposed algorithm is numerically examined by applying it to the Chen–Lorenz–Rossler and the Duffing–Van der Pol chaotic systems. Simulation results show the effectiveness of the proposed scheme.     

State-Feedback Stabilization of Well-Posed Linear Systems

A finite-dimensional linear time-invariant system is output-stabilizable if and only if it satisfies the finite cost condition, i.e., if for each initial state there exists at least one L² input that produces an L² output. It is exponentially stabilizable if and only if for each initial state there exists at least one L² input that produces an L² state trajectory. We extend these results to well-posed linear systems with infinite-dimensional input, state and output spaces. Our main contribution is the fact that the stabilizing state feedback is well posed, i.e., the map from an exogenous input (or disturbance) to the feedback, state and output signals is continuous in L_loc² in both open-loop and closed-loop settings. The state feedback can be chosen in such a way that it also stabilizes the I/O map and induces a (quasi) right coprime factorization of the original transfer function. The solution of the LQR problem has these properties.     

Steiner triple systems of order 19 associated with a certain type of projective plane of order 10

It is shown that the existence of a Steiner triple system of order 19 satisfying certain very restrictive conditions would lead to the completion of a large portion of the incidence matrix of a projective plane of order 10.     

Multidimensional constant linear systems

A continuous resp. discrete r-dimensional (r1) system is the solution space of a system of linear partial differential resp. difference equations with constant coefficients for a vector of functions or distributions in r variables resp. of r-fold indexed sequences. Although such linear systems, both multidimensional and multivariable, have been used and studied in analysis and algebra for a long time, for instance by Ehrenpreis et al. thirty years ago, these systems have only recently been recognized as objects of special significance for system theory and for technical applications. Their introduction in this context in the discrete one-dimensional (r=1) case is due to J. C. Willems. The main duality theorem of this paper establishes a categorical duality between these multidimensional systems and finitely generated modules over the polynomial algebra in r indeterminates by making use of deep results in the areas of partial differential equations, several complex variables and algebra. This duality theorem makes many notions and theorems from algebra available for system theoretic considerations. This strategy is pursued here in several directions and is similar to the use of polynomial algebra in the standard one-dimensional theory, but mathematically more difficult. The following subjects are treated: input-output structures of systems and their transfer matrix, signal flow spaces and graphs of systems and block diagrams, transfer equivalence and (minimal) realizations, controllability and observability, rank singularities and their connection with the integral respresentation theorem, invertible systems, the constructive solution of the Cauchy problem and convolutional transfer operators for discrete systems. Several constructions on the basis of the Gröbner basis algorithms are executed. The connections with other approaches to multidimensional systems are established as far as possible (to the author).Partially supported by US Air Force Grant AFOSR-87-0249 and by Office of Naval Research Grant N 00014-86-K-0538 through the Center for Mathematical System Theory, University of Florida, Gainesville, Florida, U.S.A.     

Infinite dimensional time varying systems with nonlinear output feedback

This paper deals with time-varying systems on Banach-spaces with unbounded input and output operators and considers nonlinear dynamical perturbations of the output feedback type. We develop an analysis which would establish existence conditions for the perturbed equations and investigate the robustness of stability for this wide class of perturbations.     

Perturbation theory for nonlinear feedback control systems and spencer-goldschmidt integrability of linear partial differential equations

This paper aims to develop the differential-geometric and Lie-theoretic foundations of perturbation theory for control systems, extending the classical methods of Poincaré from the differential equation-dynamical system level where they are traditionally considered, to the situation where the element of control is added. It will be guided by general geometric principles of the theory of differential systems, seeking approximate solutions of the feedback linearization equations for nonlinear affine control systems. In this study, certain algebraic problems of compatibility of prolonged differential systems are encountered. The methods developed by D. C. Spencer and H. Goldschmidt for studying over-determined systems of partial differential equations are needed. Work in the direction of applying theio theory is presented.Supported by grants from the Ames Research Center of NASA and the Applied Mathematics and Systems Research Programs of the National Science Foundation     

Global asymptotic stability of stochastic BAM neural networks with distributed delays and reaction-diffusion terms

This paper is concerned with global asymptotic stability of a class of reaction-diffusion stochastic Bi-directional Associative Memory (BAM) neural networks with discrete and distributed delays. Based on suitable assumptions, we apply the linear matrix inequality (LMI) method to propose some new sufficient stability conditions for reaction-diffusion stochastic BAM neural networks with discrete and distributed delays. The obtained results are easy to check and improve upon the existing stability results. An example is also given to demonstrate the effectiveness of the obtained results.     

On the stability of infinite-dimensional linear inequality systems

The principal aim of this paper is to study the stability of the solution set mapping of a system composed by an arbitrary set of linear inequalities in an infinite-dimensional space. The unknowns space is assumed to be metrizable, which allows us to measure the size of any possible perturbation. Conditions guaranteeing the closedness, the lower semicontinuity and the upper semicontmuity of this mapping, at a particular nominal system, are given in the paper. The more significant differences with respect to the finite dimensional case, previously approached in the context of the so-called semi-infinite optimization, are illustrated by means of convenient examples.     

On strong regular stabilizability for linear neutral type systems

The problem of strong stabilizability of linear systems of neutral type is investigated. We are interested in the case when the system has an infinite sequence of eigenvalues with vanishing real parts. This is the case when the main part of the neutral equation is not assumed to be stable in the classical sense. We discuss the notion of regular strong stabilizability and present an approach to stabilize the system by regular linear controls. The method covers the case of multivariable control and is essentially based on the idea of infinite-dimensional pole assignment proposed in [G.M. Sklyar, A.V. Rezounenko, A theorem on the strong asymptotic stability and determination of stabilizing controls, C. R. Acad. Sci. Paris Ser. I Math. 333 (8) (2001) 807-812]. Our approach is based on the recent results on the Riesz basis of invariant finite-dimensional subspaces and strong stability for neutral type systems presented in [R. Rabah, G.M. Sklyar, A.V. Rezounenko, Stability analysis of neutral type systems in Hilbert space, J. Differential Equations 214 (2) (2005) 391-428].     

Finite L 2-gain with nondifferentiable storage functions

We consider affine control systems with the finite L₂-gain property in the case the storage function is nondifferentiable. We generalize some classical results concerning the connection of the finite L₂-gain property with the stability properties of the unforced system, the characterization of finite L₂-gain by means of partial differential inequalities of the Hamilton-Jacobi type and the problem of giving to a system the finite L₂-gain property by means of a feedback law. Moreover, we introduce and study the apparently new notion of exact storage function.     

Existence and Stability of Periodic Solutions of a Duffing Equation by Using a New Maximum Principle

The purpose of this paper is to obtain new criteria for existence and asymptotic stability of periodic solutions of a Duffing equation x + cx + g(t, x) = 0, taking advantage of a new maximum principle with L^p-conditions combined with known relations between upper and lower solutions, topological degree and stability.Partially supported by D.G.I. BFM2002-01308, Ministerio Ciencia y Tecnología, Spain.     

Nonlinear feedback control and systems of partial differential equations

Finding an equivalence between two feedback control systems is treated as a problem in the theory of partial differential equation systems. The mathematical aim is to embed the Jakubzyk-Respondek, Hunt-Meyer-Su work on feedback linearization in the general theory of differential systems due to Lie, Cartan, Vessiot, Spencer, and Goldschmidt. We do this by using the functor taking control systems into differential systems, and studying the equivalence invariants of such differential systems. After discussing the general case, attention is focussed on the special situation of most immediate practical importance, the theory of feedback linearization. In this case, the general system for feedback equivalence becomes a system of linear partial differential equations. Conditions are found that the general solution of this system may be described in terms of a Frobenius system and certain differential-algebraic operations.This work was supported by grant from the Ames Research Center of NASA and the Applied Mathematics Program of the National Science Foundation.     

New filter design for linear time-delay systems

This paper develops new robust delay-dependent filter design for a class of linear systems with time-varying delays and convex-bounded parameter uncertainties. The design procedure hinges upon the constructive use of an appropriate Lyapunov functional plus a free-weighting matrices in order to exhibit the delay-dependent dynamics. The developed approach utilizes smaller number of LMI decision variables thereby leading to less conservative solutions to the delay-dependent stability and filtering problems. Subsequently, linear matrix inequalities (LMIs)-based conditions are characterized such that the linear delay system is robustly asymptotically stable with an γ-level L₂-gain. All the developed results are tested on representative examples.     

Bitangential interpolation for input-output maps of time-varying systems: The continuous time case

This paper introduces and presents the solution of the time-varying analogue of the two-sided Nudelman interpolation problem. As a first step the related Lagrange-Sylvester interpolation problem is solved.