Stability in systems with unbounded aftereffect

In this paper, we examine the problem of the stability analysis for linear delay-differential systems. Using Lyapunov method, we present sufficient conditions for the stability of the systems in terms of linear matrix inequality (LMI) Based on the Lyapunov– Krasovskii functional techniques which can be easily solved by using YALMIP Tool box. Numerical examples are given to illustrate our results.

2014

Abstract. It is shown that every solution of a linear differential system with constant coefficients and time delays tends to zero if a certain matrix derived from the coefficient matrix is a nonsingular M-matrix and the diagonal delays satisfy the so-called 3/2 condition.

intechopen., 2018

This chapter presents an extension and offers a more comprehensive overview of our previous paper entitled "Stability conditions for a class of nonlinear time delay systems" published in "Nonlinear Dynamics and Systems Theory" journal. We first introduce a more complete approach of the nonlinear system stability for the single delay case. Then, we show the application of the obtained results to delayed Lur'e Postnikov systems. A state space representation of the class of system under consideration is used and a new transformation is carried out to represent the system, with delay, by an arrow form matrix. Taking advantage of this representation and applying the Kotelyanski lemma in combination with properties of M-matrices, some new sufficient stability conditions are determined. Finally, illustrative example is provided to show the easiness of using the given stability conditions.

SIAM Journal on Mathematical Analysis, 2002

It is shown that every solution of a linear differential system with constant coefficients and time delays tends to zero if a certain matrix derived from the coefficient matrix is a nonsingular M-matrix and the diagonal delays satisfy the so-called 3/2 condition.

Journal of Applied and Industrial Mathematics, 2012

This paper introduces some sufficient conditions for uniform and asymptotic global stability as well as the algorithms for design of stabilizing control for special systems like cascaded (triangular) systems and integrator chains. The results are presented in terms of semidefinite Lyapunov functions, and they hold for nonlinear nonautonomous systems. Application of the results proposed is illustrated by some classical examples.

Russian Mathematics, 2008

We study certain sufficient conditions for the local and global uniform asymptotic stability, as well as the stabilizability of the equilibrium in cascade systems of delay differential equations. As distinct from the known results, the assertions presented in this paper are also valid for the cases, when the right-hand sides of equations are nonlinear and depend on time or arbitrarily depend on the historical data of the system. We prove that the use of auxiliary constant-sign functionals and functions with constant-sign derivatives essentially simplifies the statement of sufficient conditions for the asymptotic stability of a cascade. We adduce an example which illustrates the use of the obtained results. It demonstrates that the proposed procedure makes the study of the asymptotic stability and the construction of a stabilizing control easier in comparison with the traditional methods.

IEEE Transactions on Automatic Control, 2015

General nonlinear differential systems with time-varying delay are considered. Several explicit criteria for global exponential stability are presented. Two examples are given to illustrate the obtained results.

This paper addresses exponential stability problem for a class of linear systems with time delay. By constructing a suitable augmented Lyapunov-Krasovskii functional combined with Leibniz-Newton's formula, new sufficient conditions for the exponential stability of the systems are first established in terms of LMIs.

The article provides sufficient conditions for both practical and finite time stability of linear continuous time delay systems described as

Taiwanese Journal of Mathematics, 2014

General linear time-varying differential systems with delay are considered. Several explicit criteria for exponential stability are presented. Furthermore, an explicit robust stability bound for systems subject to time-varying perturbations is given. Two examples are given to illustrate the obtained results. To the best of our knowledge, the results of this paper are new.

Journal of Mathematical Analysis and Applications, 1998

We consider a class of linear delay equations with perturbed time lags and present conditions which guarantee that the asymptotic stability of the trivial solution of the equation at hand is preserved under these perturbations. As an example we show how our results can be used to obtain an estimate on the maximum allowable sampling interval in the stabilization of a hybrid system with feedback delays. We also present applications of our perturbation theorem to obtain stability conditions for delay equations with multiple delays. ᮊ 1998 Academic Press *

Abstract and Applied Analysis, 2011

New explicit conditions of asymptotic and exponential stability are obtained for the scalar nonautonomous linear delay differential equationx˙(t)+∑k=1mak(t)x(hk(t))=0with measurable delays and coefficients. These results are compared to known stability tests.

IFAC Proceedings Volumes, 1998

In this paper, some recent stability results on linear time-delay systems are outlined. The goal is to give an overview of the state of the art of the techniques used in delay system stability analysis. In particular, two specific problems (delay-independent / delay-dependent) are considered and some references where the reader can find more details and proofs are pointed out. This paper is based on Niculescu et al. (1997).

Proceedings of the American Control Conference, 2004

In this paper, we extend the concepts of dissipativity and exponential dissipativity to provide new sufficient conditions for guaranteeing asymptotic stability of a time delay dynamical system. Specifically, representing a time delay dynamical system as a negative feedback interconnection of a finite-dimensional linear dynamical system and an infinitedimensional time delay operator, we show that the time delay operator is dissipative. As a special case of this result we show that the storage functional of the dissipative delay operator involves an integral term identical to the integral term appearing in standard Lyapunov-Krasovskii functionals. Finally, using stability of feedback interconnection results for dissipative systems, we develop new sufficient conditions for asymptotic stability of time delay dynamical systems. The overall approach provides an explicit framework for constructing Lyapunov-Krasovskii functionals as well as deriving new sufficient conditions for stability analysis of asymptotically stable time delay dynamical systems based on the dissipativity properties of the time delay operator.

2005

In this paper the concepts of dissipativity and the exponential dissipativity are used to provide sufficient conditions for guaranteeing asymptotic stability of a time delay dynamical system. Specifically, representing a time delay dynamical system as a negative feedback interconnection of a finite-dimensional linear dynamical system and an infinite-dimensional time delay operator, we show that the time delay operator is dissipative with respect to a quadratic supply rate and with a storage functional involving an integral term identical to the integral term appearing in standard Lyapunov-Krasovskii functionals. Finally, using stability of feedback interconnection results for dissipative systems, we develop sufficient conditions for asymptotic stability of time delay dynamical systems. The overall approach provides a dissipativity theoretic interpretation of Lyapunov-Krasovskii functionals for asymptotically stable dynamical systems with arbitrary time delay.