Integral inequalities, stability analysis, continuoustime delay systems, hierarchy of lmis. Integral inequalities and stability of an equilibrium. Two general integral inequalities and their applications to stability. This paper focuses on stability analysis for a class of genetic regulatory networks with interval timevarying delays. Combining advanced integral inequalities and slack variables, a new delaydependent. T1 stability analysis with dissipation inequalities and integral quadratic constraints. A generalized double integral inequalities approach to.
Then, based on the proposed inequality, we derive a stability criterion for the system. Gronwallbellman type integral inequalities and applications to global uniform asymptotic stability mekki hammi and mohamed ali hammami university of sfax, faculty of sciences of sfax, department of mathematics, route soukra, bp 1171, 3000 sfax, tunisia, mohamedali. Combining advanced integral inequalities and slack variables, a new delaydependent stability condition for time delay systems is. Pdf some new integral inequalities and their applications. Integral inequalities play an essential role in the stability analysis of continuoustime systems with time delay. This paper is concerned with the problem of delaydependent stability of timedelay systems. Firstly, it introduces a new useful integral inequality which has been proved to be less conservative than the previous inequalities. Quantum integral inequalities on finite intervals journal. Exponential stability of timedelay systems via new weighted.
Multiple integral inequalities and stability analysis of time delay. Martynyuk soviet applied mechanics volume 16, pages 267 281 1980 cite this article. As applications, based on these new inequalities, some sufficient conditions for global uniform asymptotic stability are obtained. The stability analysis of fractionalorder systems with unbounded delay remains an open problem. Z p 0 f s ds p c 00 r tions x is the banach space endowed with the sup norm of all continuous func. A characterization of integral inputtostate stability. Pdf multipleintegral inequalities to stability analysis of.
Dec 01, 2016 read two general integral inequalities and their applications to stability analysis for systems with time. In order to make full use of the delay information, a novel lyapunovkrasovskii. The study of the solution of a fredholmvolterra integral. Two general integral inequalities and their applications. In this paper, we study nonlinear fractional differential equations with hadamard derivative and ulam stability in the weighted space of continuous functions. This paper investigates the new stability criteria for the asymptotic stability of timedelay systems via integral inequalities and jensen inequalities. Nov 18, 2018 recently, the jensen integral inequality was successfully generalized to the bessel. Stability analysis of systems with timevarying delay via. Introduction one of the main issues in control design concerns the. Checking a certain integral inequality for given system parameters, or alternatively. Keywords asymptotic stability integral inequality fractional halanay inequality unbounded timedelay b. The results concern stability in the sense of liapunov. The inputoutput behavior of the perturbation is described by an integral quadratic constraint iqc.
This correspondence is concerned with the stability for interval timedelay systems. By virtue of the newly derived inequalities, the resulting criteria are less conservative than some existing literature. Bessel summation inequalities for stability analysis of. Then, we use the improved integral inequality to deal with the resultant double integral items in the derivative of the involved lyapunovkrasovskii functional. This paper is concerned with the robust stability analysis for uncertain systems with interval timevarying delay. Finding integral inequalities for quadratic functions plays a key role in the field of stability analysis. Our aim in this paper is to investigate some delay integral inequalities on time scales by using gronwalls inequality and comparison theorem. In this chapter, we will focus on the study of integral inequalities. First, a new augmented lyapunovkrasovskii functional lkf is constructed, which includes three integral terms in the form of. Using these two integral inequalities obtained, the halanay inequality with unbounded delay is extended to caputo fractionalorder case and riemannliouville fractionalorder case.
Stability analysis of systems with interval timevarying. New stability criteria for asymptotic stability of timedelay. Read two general integral inequalities and their applications to stability analysis for systems with time. Integral inequalities have been widely used in stability analysis for systems with timevarying delay because they directly produce bounds for integral terms. Based on a new integral inequality and a generalized reciprocally convex combination matrix inequality, a new delaydependent stability criterion is obtained in terms of a linear matrix inequality lmi.
In this paper, new weighted integral inequalities wiis have been proposed. Gronwallbellman type integral inequalities and applications. In this paper, we firstly explore two new integral inequalities. Multiple integral inequalities and stability analysis of time. An integral inequality in the stability problem of timedelay systems abstract. New integral inequalities and asymptotic stability of fractionalorder systems with unbounded time delay. Finally, two numerical examples are given to illustrate the effectiveness of the present result. In 24, by using wirtingerbased single and double integral inequalities and delay decomposition technique, the stability analysis of neural systems is investigated. Secondly, the new delaydependent lyapunovkrasovskii functional based on the double integral inequalities and. Multiple integral inequalities and stability analysis of. An integral inequality is derived, and applied to the stability problem of timedelay systems using discretized lyapunov functional formulation. Jun 30, 2018 the stability analysis of fractionalorder systems with unbounded delay remains an open problem. Integral inequalities and stability of an equilibrium state on a time scale. Then, to show the effectiveness of the new integral inequality, a new delaydependent stability criterion is derived in terms of linear matrix inequality lmi.
A new multiple integral inequality and its application to. These results include many existing ones as special cases and can be used as tools in the qualitative analysis of certain classes of integrodifferential. Iqcs provide sufficient conditions for the stability of complicated interconnected systems, and these conditions can be checked by semidefinite programming. New stability criteria for asymptotic stability of time. This paper presents two general integral inequalities from which almost all of the existing integral inequalities can be obtained, such as jensen inequality, the wirtinger. Finally, the merits of the proposed criterion are shown by two numerical. These inequalities can be used in the study of stability problems of perturbed dynamic systems. Some integral inequalities with maximum of the unknown functions snezhana g. We develop a new integral inequality which is proved to encompasses the. Pdf improved integral inequalities for stability analysis. Mar 26, 2014 in this paper, some of the most important integral inequalities of analysis are extended to quantum calculus. The developed inequality has been applied to access the stability of a linear retarded system to estimate a maximum delay upperbound. Pdf this paper considers the stability analysis of timevarying delay systems.
Jan 01, 2019 second, indepth analysis of the bessellegendre inequality and some affine integral inequalities is made, and recent stability results are also summarised, including stability criteria for three cases of a timevarying delay, where information on the bounds of the timevarying delay and its derivative is totally known, partly known and. The paper deals with some results of the application of the method of integral inequalities to problems of qualitative analysis, involving the stability of unperturbed motion. Jensens inequality convex functions and a proof for. These include the holder, hermitehadamard, trapezoid, ostrowski, cauchybunyakovskyschwarz, gruss, and grusscebysev integral inequalities. Two novel integral inequalities are developed in this paper for this estimation task. In this paper, the stability analysis for timedelay systems is investigated. This paper develops a new framework to analyze and design iterative optimization algorithms built on the notion of integral quadratic constraints iqcs from robust control theory. Using frequency domain method, the asymptotic stability resultsaboutfractionalretardedandneutralsystemsare obtained in 24,25. Firstly, some new nonlinear integral inequalities with hadamard type singular kernel are established, which can be used in the theory of certain classes of fractional differential equations. The wirtingerbased integral inequality 9 and the auxiliary function. A new integral inequality and application to stability of. Recently, the jensen integral inequality was successfully generalized to the bessel.
The gronwallbellman inequality and its various linear and nonlinear generalizations are crucial in the discussion of the existence, uniqueness, continuation, boundedness, oscillation, stability and other qualitative. Various types of integral inequalities are solved in the papers 1, 7, 9, 15. Triple integral inequalities without slack variables are proposed, which show more and more evident improvements as the degrees of intermediate polynomials increase. Based on the generalized halanay inequality, we establish several asymptotical stability conditions under which the fractional order systems with time delay are asymptotically stable.
We define an l2 function by its fourier coefficients. Pdf multipleintegral inequalities to stability analysis. Overview of recent advances in stability of linear systems. Applications from this books results can be found in many areas of pure and applied mathematics, especially in ordinary and partial differential equations and. In this paper we will use the picard operators technique, in order to establish the existence and uniqueness, data dependence and gronwalltype results for the solutions of a fredholmvolterra functional integral equation. This study is concerned with the stability of a linear system with an interval timevarying delay. An integral inequality in the stability problem of time. Based on the augmented lkf with triple integral terms, their applications to stability analysis for interval timedelay systems are provided. In this paper, some new nonlinear integral inequalities are established.
Secondly, the new delaydependent lyapunovkrasovskii functional based on the double integral inequalities. Frontiers in functional equations and analytic inequalities. A novel stability analysis for the interval timedelay systems is proposed by employing a new series of integral inequalities for single and double integrals. Firstly, further improved double integral inequalities in the forms of infinite series are developed for estimation of the derivatives of triple integral terms.
Stability analysis for systems with time delays via new. An improved integral inequality concerning on double integral items is first established. Aug 30, 2019 integral inequalities play an essential role in the stability analysis of continuoustime systems with time delay. Furthermore, a mathematical relationship of the new double ii with existing inequalities is discussed to show that the developed inequality is more general, effective and bears less computational burden. Also we show applications of these inequalities for. This paper is devoted to stability analysis of continuoustime delay systems based on a set of lyapunovkrasovskii functionals. Several new retarded integral inequalities of gronwallbellmanpachpatte type are presented which generalize the inequalities to the more general nonlinear case in the literature and provide explicit bounds on unknown functions. Different from the recently introduced wirtingerbased inequalities, refined jensen inequalities and auxiliary functionbased inequalities, the proposed ones can provide more accurate bounds for the cross terms in. Firstly, based on an integral equality, a new integral inequality is obtained. New integral inequalities and asymptotic stability of. Analysis and design of optimization algorithms via integral. Some new integral inequalities and their applications in studying the stability of nonlinear integrodifferential equations with time delay. Various types of linear and nonlinear integral inequalities are presented in a systematic manner, starting with the gronwall inequality and including inequalities for interval functions intended.
N2 this technical note considers the stability of a feedback connection of a known linear, timeinvariant system and a perturbation. Finally, the merits of the proposed criterion are shown by two numerical examples. Some generalized nonlinear gamidov type integral inequalities. Improved integral inequalities for stability analysis of. Exponential stability of timedelay systems via new. Stability analysis with dissipation inequalities and.
Recent developments of integral inequalities and its. In 18,26, the authors explored the fractionalorder systems with unbounded delay and they obtained the stability of the systems with the restriction. Improved integral inequalities for stability analysis of interval time. Asymptotical stability of fractional order systems with. The paper deals with the stability analysis of the following continuoustime delay system 1. The freeweighting matrix and integral inequality methods are widely used to derive delaydependent criteria for the stability analysis of timevaryingdelay systems because they avoid both the use of a model transformation and the. He college of information science and technology, donghua university, shanghai 201620, peoples republic of china. The paper ends with a result of the ulamhyers stability of this integral equation. Integral inequalities have been widely used in stability analysis for systems with timevarying delay because they directly produce bounds for integral terms with respect to quadratic functions. Next, the inequality combines delaydecomposition approach with uncertain parameters applied to timedelay systems, based on the new lyapunovkrasovskii functionals and. Pdf freematrixbased integral inequality for stability analysis of.
By employing wiis, new exponential stability criteria have been derived for some classes of timedelay systems in the framework of linear matrix inequalities. Freematrixbased integral inequality for stability analysis of systems with timevarying delay abstract. Based on the augmented lkf with tripleintegral terms, their applications to stability analysis for interval timedelay systems are provided. Pdf integral inequality for timevarying delay systems. Request pdf an integral inequality in the stability problem of timedelay systems an integral inequality is derived, and applied to the stability problem of timedelay systems using. Asymptotic stability and integral inequalities 349 here lp r x x denotes the banach space of all bochner measurable functions f. Two general integral inequalities and their applications to. So, developing an accurate integral inequality is of particular importance. There are examples of functions for which the left side of 4 is finite while the right side is infinite. Applications to the investigation of the properties of solutions of various classes of equations such as uniqueness, stability, dichotomy, asymptotic equivalence and behaviour is also discussed. A new integral inequality and delaydecomposition with.
New multiple integral inequalities are derived that involve the. We show that the newly derived inequalities in this paper encompass both the. It was also shown that general inequality contributes to the less conservatism of stability criteria. Method of integral inequalities in the theory of stability of motion a. This paper investigates the stability of linear systems with a timevarying delay.
Combining advanced integral inequalities and slack variables, a new delaydependent stability condition for time delay systems is developed in 25. This paper focuses on the stability problem of linear timedelay systems. Secondly, based on the multiple integral inequality, a new delaydependent stability criterion is derived for timedelay systems. Nonlinear integral inequalities and applications to. Method of integral inequalities in the theory of stability. Index terms dissipation inequalities, finite gain, inputtostate stability, nonlinear systems, tracking. Chapter i and ii consider classical linear and nonlinear integral inequalities. This paper focuses on delaydependent stability analysis for systems with interval timevarying delays. Stability analysis with dissipation inequalities and integral quadratic constraints peter seiler abstract this paper considers the stability of a feedback connection of a known linear, timeinvariant system and a perturbation. The problem of stability analysis of nonlinear timevarying systems has attracted the attention of several researchers and has produced a vast body of important. New multiple integral inequalities are derived that involve the famous jensens and wirtingers inequalities, as well as the recently presented bessellegendre inequalities of seuret and gouaisbaut 2015 and the wirtingerbased multipleintegral inequalities. Motion stability method of integral inequalities nasaads. Some integral inequalities with maximum of the unknown functions. New multiple integral inequalities are derived that involve the famous jensens and wirtingers inequalities, as well as the recently presented bessellegendre inequalities of a.
Generalized double integral inequalities and their. When the underlying pde system is autonomous, the integral inequalities obtained in all aforementioned applications depend on time only through the state wt,x, and since they are imposed pointwise in time, the time dependence of wcan be dropped. Supremum by integral inequalities european journal of pure and. Some integral inequalities for functions of two variables. New double integral inequality with application to stability. This paper will develop two new inequalities for the above estimation task. Recent developments of integral inequalities and its applications research in the fascinating area of integral inequalities, including several surprising applications and new directions, has been described in a great number of papers and monographs. The inequalities are also in the environments of fractional calculus and conformable fractional calculus. These keywords were added by machine and not by the authors. Firstly, not only the known constant time delay, but also the unknown timevarying delay is considered for the linear system. Summary integral inequalities have been widely used in stability analysis for systems with time. Jul 03, 20 read generalized double integral inequalities and their applications in studying the stability of nonlinear integrodifferential systems with time delay, journal of dynamical and control systems on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips.
Our results unify and extend some delay integral inequalities and. The analysis relies on the notions of qderivative and qintegral on finite intervals introduced by the authors in tariboon and ntouyas in. Pdf the freeweighting matrix and integralinequality methods are widely used to derive delaydependent criteria for the stability analysis of. An improved integral inequality to stability analysis of. Stability analysis for systems with time delays via new integral inequalities abstract. Firstly, a generalized vectorsbased multiple integral inequality gvmii is presented, which can treat some existing results as its special cases.
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