Double Integral Inequality Research Articles

Some inequalities for double integrals and applications for cubature formula

Abstract We establish two Ostrowski type inequalities for double integrals of second order partial derivable functions which are bounded. Then, we deduce some inequalities of Hermite-Hadamard type for double integrals of functions whose partial derivatives in absolute value are convex on the co-ordinates on rectangle from the plane. Finally, some applications in Numerical Analysis in connection with cubature formula are given.

Synchronization control of Markov jump neural networks with mixed time-varying delay and parameter uncertain based on sample point controller

This paper put forward an improved synchronization problem for neural networks with Markov jump parameters. The traditional Markov jump neural network (MJNN) only considers the basic external time-varying delays, ignoring both the distributed and leakage delays in the internal transmission of the neural network and the small time-varying errors in the mode switching of Markov probability transition rates. In this paper, we focus on the synchronization of MJNN with mixed time-varying delay. And an improved Lyapunov–Krasovskii functional is constructed. The convergence of inequalities is solved by using affine Bessel–Legendre inequalities and Wirtinger double integral inequalities. At the same time, a new method is used to optimize the mathematical geometric area of the time-varying delay and reduce the conservativeness of the system. Finally, a sample point controller is constructed to synchronize the driving system and the corresponding system.

New result on delay-dependent stability for Markovian jump time-delay systems with partial information on transition probabilities

This paper focuses on the delay-dependent stability for a kind of Markovian jump time-delay systems &#x0028 MJTDSs &#x0029, whose transition rates are incompletely known. In order to reduce the computational complexity and achieve better performance, auxiliary function-based double integral inequality is combined with extended Wirtinger&#x02BC s inequality and Jensen inequality to deal with the double integral and the triple integral in augmented Lyapunov-Krasovskii function &#x0028 ALKF &#x0029 and their weak infinitesimal generator respectively, the more accurate approximation bounds with a fewer variables are derived. As a result, less conservative stability criteria are proposed in this paper. Finally, numerical examples are given to show the effectiveness and the merits of the proposed method.

Stability analysis for linear time-delay systems using new inequality based on the second-order derivative

This paper studies the stability problem of linear time-varying delay system. Firstly, a double integral inequality based on the second-order derivative is proposed in this paper. Secondly, novel Lyapunov–Krasovskii functional consisting of integral terms based on the second-order derivative is constructed to enhance the feasible region of delay-dependent stability. Based on the two aspects, new delay-dependent stability criteria which guarantee the asymptotic stability of linear systems with time-varying delay are given in the form of linear matrix inequality (LMI). Finally, several numerical examples are given to show the advantages of the proposed methods.

Finite-time extended dissipativity of delayed Takagi–Sugeno fuzzy neural networks using a free-matrix-based double integral inequality

This study focuses on the finite-time extended dissipativity of delayed Takagi–Sugeno (T–S) fuzzy neural networks (NNs). Based on the concept of extended dissipativity, this paper solves the $$H_\infty$$ , $$L_2-L_\infty$$ , passive, and $$({{\mathcal {Q}}}, {{\mathcal {S}}}, {\mathcal {R}})$$ -dissipativity performance in a unified framework. Using the free-matrix-based double integral inequality and an extended Wirtinger inequality in the Lyapunov–Krasovskii functional, sufficient conditions are derived to guarantee that the considered NNs are finite-time bounded, whereupon the finite-time extended dissipativity criteria for delayed T–S fuzzy NNs are constructed. The derived conditions guarantee the extended dissipativity and stability of the NNs. Three numerical examples are given to demonstrate the reduced conservatism and the effectiveness of the obtained results.

On some explicit bounds of integral inequalities related to time scales

In this paper, we propose a new method to unify and extend some nonlinear dynamic double integral inequalities of two independent variables involving pairs of time scales via nabla derivative. The acquired results give explicit bounds which can be utilized to consider the subjective and quantitative properties of specific classes of dynamic conditions on time scales. An application to prove the validity of our established results is also given.

Stability and dissipativity analysis for uncertain Markovian jump systems with random delays via new approach

ABSTRACTThis paper studies the problem of stability and dissipativity analysis for uncertain Markovian jump systems (UMJSs) with random time-varying delays. Based on the auxiliary function-based integral inequality (AFBII) and with the help of some mathematical tools, a new double integral inequality (NDII) is developed. Then, to show the efficiency of the proposed inequality, a suitable Lyapunov-Krasovskii functional (LKF) is constructed with augmented delay-dependent terms. By employing integral inequalities, new delay-dependent sufficient conditions are derived in terms of linear matrix inequalities (LMIs). Finally, illustrative examples are given to show the effectiveness and less conservatism of the results.

New double integral inequality with application to stability analysis for linear retarded systems

This paper presents the development of a new double integral inequality (II) with the motivation of yielding quadratic approximation. It is well known that approximating integral quadratic terms with quadratic terms involves a certain degree of conservatism. In this paper, a sufficient gap has been identified in the approximation of two recent IIs reported in the literature, thereby leading to the new double II. The developed inequality has been applied to access the stability of a linear retarded system to estimate a maximum delay upper-bound. 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. Four numerical examples are given to validate the authors' claim with regard to the effective estimate of delay bound results for a linear retarded system.

Delayed impulsive control for exponential synchronization of stochastic reaction–diffusion neural networks with time-varying delays using general integral inequalities

The intention of this paper is to explore the exponential synchronization of delayed stochastic reaction–diffusion neural networks. By constructing appropriate Lyapunov–Krasovskii functional (LKF) with triple integral terms and by utilizing improved double integral inequalities, new synchronization criteria are developed in the form of linear matrix inequalities (LMIs) to assure the exponential synchronization of the neural networks under study. The resulting criteria use more data of the delay bounds, and by means of new inequalities, they are substantiated to be less conservative and also computationally appealing than certain existing works. At the end, numerical simulations are furnished to reveal the potency of the acquired theoretical results.

Non-fragile sampled-data stabilization analysis for linear systems with probabilistic time-varying delays

This paper deals with the problem of non-fragile sampled-data stabilization analysis for a class of linear systems with probabilistic time-varying delays via new double integral inequality approach. Based on the auxiliary function-based integral inequality (AFBII) and with the help of some mathematical approaches, a new double integral inequality (NDII) is developed. Then, to demonstrate the merits of the proposed inequality, an appropriate Lyapunov–Krasovskii functional (LKF) is constructed with some augmented delay-dependent terms. By employing integral inequalities, an enhanced stability criterion for the concerned system model is derived in terms of linear matrix inequalities (LMIs). Finally, three benchmark illustrative examples are given to validate the effectiveness and advantages of the proposed results.

New absolute stability results for Lurie systems with interval time‐varying delay based on improved Wirtinger‐type integral inequality

SummaryThis work focuses on the absolute stability problem of Lurie control system with interval time‐varying delay and sector‐bounded nonlinearity. Firstly, we present a refined Wirtinger's integral inequality and establish an improved Wirtinger‐type double integral inequality. Secondly, a modified augmented Lyapunov‐Krasovskii functional (LKF) is constructed to analyze the stability of Lurie system, where the information on the lower and upper bounds of the delay and the delay itself are fully exploited. Based on the proposed integral inequalities and some bounding techniques, the upper bound of the derivative of the LKF can be estimated more tightly. Accordingly, the proposed absolute stability criteria, formulated in terms of linear matrix inequalities, are less conservative than those in previous literature. Finally, numerical examples demonstrate the effectiveness and advantage of the proposed method.

Delay range‐and‐rate dependent stability criteria for systems with interval time‐varying delay via a quasi‐quadratic convex framework

SummaryThis paper concerns the stability analysis of systems with interval time‐varying delay. A Lyapunov‐Krasovskii functional containing an augmented quadratic term and certain triple integral terms is constructed to integrate features of the truncated Bessel‐Legendre inequality less conservative than Wirtinger inequality that encompasses Jensen inequality, respectively, and to exploit merits of the newly developed double integral inequalities tighter than auxiliary function‐based, Wirtinger, and Jensen double integral inequalities. A new quadratic convex lemma is proposed to derive delay and its derivative dependent sufficient stability conditions in terms of linear matrix inequalities synthetically with reciprocal convex approach and affine convex combination. The efficiency of the presented method is illustrated on some classical numerical examples.

Improved result on state estimation for complex dynamical networks with time varying delays and stochastic sampling via sampled-data control

This paper investigates state estimation for complex dynamical networks (CDNs) with time-varying delays by using sampled-data control. For the simplicity of technical development, only two different sampling periods are considered whose occurrence probabilities are given constants and satisfy Bernoulli distribution, which can be further extended to the case with multiple stochastic sampling periods. By applying an input-delay approach, the probabilistic sampling state estimator is transformed into a continuous time-delay system with stochastic parameters in the system matrices, where the purpose is to design a state estimator to estimate the network states through available output measurements. By constructing an appropriate Lyapunov–Krasovskii functional (LKF) containing triple and fourth integral terms and applying Wirtinger-based single and double integral inequality, Jenson integral inequality technique, delay-dependent stability conditions are established. The obtained conditions can be readily solved by using the LMI tool box in MATLAB. Finally, a numerical example is provided to demonstrate the validity of the proposed scheme.

New stability criteria for asymptotic stability of time-delay systems via integral inequalities and Jensen inequalities

This paper investigates the new stability criteria for the asymptotic stability of time-delay systems via integral inequalities and Jensen inequalities. Firstly, not only the known constant time delay, but also the unknown time-varying delay is considered for the linear system. Secondly, the new delay-dependent Lyapunov–Krasovskii functional based on the double integral inequalities and Jensen inequalities is introduced, such that the linear system with time-delay is asymptotically stable. Thirdly, two classes of delay-dependent stability conditions in terms of linear matrix inequalities (LMIs) are derived, such that the control design conditions are relaxed and computation complexity is reduced. Compared with previous works, the larger feasible solution region and less conservative results are obtained. Finally, some numerical examples are performed to show the effectiveness and advantage of the proposed method.

Consensus of multiagent systems with nonlinear dynamics and time‐varying communication delays

SummaryThis paper deals with the local consensus of multiagent systems with nonlinear dynamics communication delays simultaneously. By introducing a weighted average state and applying the properties of the Laplacian matrix eigenvalues, the system is decoupled into several subsystems, firstly, to reduce complexity of theory analysis. Then, a new augmented vector containing single and double integral terms is constructed and the corresponding Lyapunov functional with triple integral terms is introduced. Meanwhile, in order to improve the estimating accuracy of the derivatives of constructed Lyapunov functional, single integral inequalities and double integral inequalities via auxiliary functions, an extended relaxed integral inequality and an reciprocally convex approach are used, as a result, stability criterion with less conservatism is derived, which guarantees the local consensus of the considered systems. Finally, numerical examples are provided to check the improvement of the proposed method over the existing works.

Sampled data Control for Takagi-Sugeno Fuzzy Systems with Actuator Saturation

This paper deals with the problem of sampled data control for a class of Takagi-Sugeno fuzzy systems with actuator saturation. On the basis of the Lyapunov-Krasovskii functional (LKF) and Wirtinger-based double integral inequality, a set of sufficient conditions are derived in terms of linear matrix inequalities (LMIs) which can be easily solved by using some standard numerical packages, then, a state feedback controller is constructed for guaranteing the stability of the considered system. Eventually, the effectiveness of the proposed method have been validated through numerical simulations.

New Ostrowski and Ostrowski-Gruss type inequalities for double integrals on time scales involving a combination of $\Delta$-integral means

In 2014, some Ostrowski type inequalities for functions of a single variable were obtained in [Y. Jiang, H. R\"uzgar, W. J. Liu and A. Tuna: Some new generalizations of Ostrowski type inequalities on time scales involving combination of $\Delta$-integral means, J. Nonlinear Sci. Appl., {\bf{7}} (2014), 311--324]. In this paper, we extend some of the inequalities obtained in the above paper for double integrals. One of our results generalizes a result in the article [W. J. Liu, Q. A. Ng\^o and W. Chen: On new Ostrowski type inequalities for double integrals on time scales, Dyn. Syst. Appl., {\bf 19} (2010), 189--198]. We also apply our results to the continuous, discrete and quantum time scales to obtain some interesting inequalities.

The stability analysis of time-varying delayed systems based on new augmented vector method

This paper focuses on the stability analysis of systems with interval time-varying delay. A new augmented vector containing single and double integral terms is constructed and the corresponding Lyapunov functional with triple integral terms is introduced. In order to improve the estimating accuracy of the derivatives of the constructed Lyapunov functional, single integral inequalities and double integral inequalities via auxiliary functions are employed on the first step, then an extended relaxed integral inequality and reciprocally convex approach are further utilized to narrow the scaling room of the functional derivatives. As a result, some novel delay-dependent stability criteria with less conservatism are derived. Finally, numerical examples are provided to check the effectiveness of the theoretical results and the improvement of the proposed method over the existing works.

Behavior around isolated singularity for integral inequalities with multiple Riesz potentials

ABSTRACTWe are interested in the asymptotic behavior around the origin for nonnegative functions , satisfying the double integral inequality where p,q>0 and 0<a,b<n. We next apply these results to the study of the double elliptic inequality where and .

Stability Analysis for Time-delay Systems with Nonlinear Disturbances via New Generalized Integral Inequalities

This paper represents a novel less conservative stability criterion for time-delay systems with nonlinear disturbances. The main purpose is to obtain larger upper bound of the time-varying delay. A suitable Lyapunov- Krasovskii functional (LKF) with triple integral terms is constructed. Then, two new generalized double integral (GDI) inequalities are proposed which encompass Wirtinger-based double inequality as a special case. A simple case of the proposed GDI inequality is utilized to estimate double integral terms in the time derivative of the constructed LKF. Further, an improved delay-dependent stability criterion is derived in the form of linear matrix inequalities (LMIs). Finally, some numerical examples are given to illustrate the improvement of the proposed criteria.