Double Integral Inequality Research Articles

New stochastic synchronization criteria for fuzzy Markovian hybrid neural networks with random coupling strengths

This paper focuses on the stochastic synchronization problem for a class of fuzzy Markovian hybrid neural networks with random coupling strengths and mode-dependent mixed time delays in the mean square. First, a novel free-matrix-based single integral inequality and two novel free-matrix-based double integral inequalities are established. Next, by employing a novel augmented Lyapunov–Krasovskii functional with several mode-dependent matrices, applying the theory of Kronecker product of matrices, Barbalat’s Lemma and the new free-matrix-based integral inequalities, two delay-dependent conditions are established to achieve the globally stochastic synchronization for the mode-dependent fuzzy hybrid coupled neural networks. Finally, two numerical examples with simulation are provided to illustrate the effectiveness of the presented criteria.

Exponential stability and extended dissipativity criteria for generalized neural networks with interval time-varying delay signals

This paper discusses the problems of exponential stability and extended dissipativity analysis of generalized neural networks (GNNs) with time delays. A new criterion for the exponential stability and extended dissipativity of GNNs is established based on the suitable Lyapunov–Krasovskii functionals (LKFs) together with the Wirtinger single integral inequality (WSII) and Wirtinger double integral inequality (WDII) technique, and that is mixed with the reciprocally convex combination (RCC) technique. An improved exponential stability and extended dissipativity criterion for GNNs are expressed in terms of linear matrix inequalities (LMIs). The major contributions of this study are an exponential stability and extended dissipativity concept can be developed to analyze simultaneously the solutions of the exponential H∞, L2−L∞, passivity, and dissipativity performance for GNNs by selecting the weighting matrices. Finally, several interesting numerical examples are developed to verify the usefulness of the proposed results, among them one example was supported by real-life application of the benchmark problem that associates with reasonable issues under extended dissipativity performance.

Further results on dissipativity analysis for Markovian jump neural networks with randomly occurring uncertainties and leakage delays

This paper is concerned with the mixed $$H_\infty$$ and dissipativity performance for Markovian jump neural networks with time delay in the leakage term and randomly occurring uncertainties. The randomly occurring uncertainties are assumed to be mutually uncorrelated Bernoulli-distributed white noise sequences. By introducing a triple-integrable term in the Lyapunov functional, the Wirtinger-based double-integral inequality is utilized to bound the derivative of the triple-integral term and then a sufficient condition is derived to ensure that the considered neural networks to be strict $$({\mathcal {Q,S,R}}) - \gamma$$ -dissipative and passive. These conditions are presented in terms of linear matrix inequalities, which can be easily solved by using standard numerical software. Finally, numerical examples are given to show the effectiveness and the potential significance of the proposed results.

A generalized double integral inequalities approach to stability analysis for time-delay systems

In this paper, the stability analysis for time-delay systems is investigated. Firstly, further improved double integral inequalities in the forms of infinite series are developed for estimation of the derivatives of triple integral terms. Comparing with the recently introduced Wirtinger-based, refined Jensen and auxiliary function-based double integral inequalities, the estimation gaps of the proposed ones are ever-shrinking with increase of the intermediate terms without requiring any free-weighting matrix. Then, by constructing a triple integral type of Lyapunov–Krasovskii functional (LKF), some stability conditions for time-delay systems are established. By virtue of the new inequalities, the derived criteria are less conservative and more computationally attractive than some existing works. Finally, the improvement and effectiveness of the proposed approaches are demonstrated via the numerical examples.

Synchronization for memristive chaotic neural networks using Wirtinger-based multiple integral inequality

This paper investigates the synchronization problem for a class of memristive chaotic neural networks with time-varying delays. Based on te Wirtinger-based double integral inequality, two novel inequalities are proposed, which are multiple integral forms of the Wirtinger-based integral inequality. Next, by applying the reciprocally convex combination approach for high order case and a free-matrix-based inequality, novel delay-dependent conditions are established to achieve the synchronization for the memristive chaotic neural networks. The results are based on dividing the bounding of activation function into two subintervals with equal length. Finally, a numerical example is provided to demonstrate the effectiveness of the theoretical results.

Stability Analysis of Delayed Genetic Regulatory Networks via a Relaxed Double Integral Inequality

Time delay arising in a genetic regulatory network may cause the instability. This paper is concerned with the stability analysis of genetic regulatory networks with interval time-varying delays. Firstly, a relaxed double integral inequality, named as Wirtinger-type double integral inequality (WTDII), is established to estimate the double integral term appearing in the derivative of Lyapunov-Krasovskii functional with a triple integral term. And it is proved theoretically that the proposed WTDII is tighter than the widely used Jensen-based double inequality and the recently developed Wiringter-based double inequality. Then, by applying the WTDII to the stability analysis of a delayed genetic regulatory network, together with the usage of useful information of regulatory functions, several delay-range- and delay-rate-dependent (or delay-rate-independent) criteria are derived in terms of linear matrix inequalities. Finally, an example is carried out to verify the effectiveness of the proposed method and also to show the advantages of the established stability criteria through the comparison with some literature.

On weighted Grüss type inequalities for double integrals

In this study, we obtain some new weighted inequalities of Grüss type for functions of two independent variables. Special cases of the results presented here reduce the results given in earlier works

Global exponential stability and dissipativity of generalized neural networks with time-varying delay signals

This paper investigates the problems of exponential stability and dissipativity of generalized neural networks (GNNs) with time-varying delay signals. By constructing a novel Lyapunov–Krasovskii functionals (LKFs) with triple integral terms that contain more advantages of the state vectors of the neural networks, and the upper bound on the time-varying delay signals are formulated. We employ a new integral inequality technique (IIT), free-matrix-based (FMB) integral inequality approach, and Wirtinger double integral inequality (WDII) technique together with the reciprocally convex combination (RCC) approach to bound the time derivative of the LKFs. An improved exponential stability and strictly (Q,S,R)-γ-dissipative conditions of the addressed systems are represented by the linear matrix inequalities (LMIs). Finally, four interesting numerical examples are developed to verify the usefulness of the proposed method with a practical application to a biological network.

Multiple-integral inequalities to stability analysis of linear time-delay systems

This paper is concerned with the stability analysis of linear systems with constant delay. First, with the help of Schmidt orthogonalization, we define a new set of orthogonal polynomials. By using the orthogonal polynomial set, we propose a novel multiple-integral inequality, which can achieve less conservatism than many existing inequalities, such as Jensen׳s single-integral inequality, Jensen׳s double-integral inequality, the Wirtinger-based single-integral inequality and the auxiliary function-based double-integral inequality. Then, based on the proposed inequality, we derive a stability criterion for the system under consideration, which is less conservative than the existing ones. Finally, we provide a numerical example to illustrate the effectiveness of the derived criterion.

Vibration control of structural systems via robust non-fragile sampled-data control scheme

This paper investigates the dissipative non-fragile output feedback sampled-data control problem for uncertain structural systems. In the proposed system, uncertainties are assumed to be time-varying and bounded, which are considered in the linear fractional transformation form. The main objective of this paper is to design a non-fragile sampled-data controller such that the resulting closed-loop system is strictly (Q,S,R)-α-dissipative. On the basis of a suitable Lyapunov–Krasovskii functional and linear matrix inequality technique, a new set of sufficient conditions is established to achieve the required result for both nominal and uncertain systems. In particular, an output feedback dissipative sampled-data controller is designed by solving a set of matrix inequalities. More precisely, Schur complement lemma, free weighting matrix approach and Wirtinger-based double integral inequality are utilized to substantially simplify the derivation of the main results. Finally, simulations based on structural systems are conducted to illustrate the effectiveness of the proposed control scheme.

Improved delay-dependent stability criteria for neutral systems with mixed interval time-varying delays and nonlinear disturbances

It is well-known that the stability analysis of time-delay systems is a key step to design appropriate controllers and/or filters for those systems. In this paper, the problem of the delay-dependent stability analysis of neutral systems with mixed interval time-varying delays with/without nonlinear perturbations is revisited. Bounded derivatives of the discrete and neutral delays with upper-bounds not limited to be strictly less than one are considered. New stability criteria are developed using the Lyapunov Krasovskii methodology which are expressed in terms of linear matrix inequalities (LMIs). An augmented Lyapunov Krasovskii functional (LKF) utilizing triple integral terms and the descriptor transformation is employed to this aim. In addition, advanced techniques such as Wirtinger-based single and double-integral inequalities, delay decomposition technique combined with the reciprocally convex approach, as well as a few effective free-weighting matrices are employed to achieve less conservative stability conditions. Comprehensive benchmarking numerical examples and simulation studies demonstrate the effectiveness of the proposed stability criteria with respect to some recently published results. The efficacy of the modern integral inequalities are also emphasized against the conventional Jensen׳s inequalities.

State feedback synchronization control of coronary artery chaos system with interval time-varying delay

In the study on synchronization problems of coronary artery chaos system, one of the often overlooked parts is the controller input delay, also mentioned as the drug absorption time during clinical trials. With the time delay considered in system, mentioned defect might cause random behavior or crashes. This paper studied the state feedback synchronization control for uncertain coronary artery chaotic systems under interval time-varying delay. The triple integral forms are added to Lyapunov–Krasovakii function to provide more information on the input delay. A single integral inequality and a double integral inequality of Wirtinger-based inequality are applied to reduce conservation for their tighter upper bound comparing with the traditional methods. What’s more, the delay interval is decomposed into multiple equidistant subintervals to take a further step of conservation reduction. With the method, the abnormal chaotic motion can be synchronized with the healthy system under instable periodic orbit so as to achieve an effective treatment.

A new double integral inequality and application to stability test for time-delay systems

This paper is concerned with stability analysis for linear systems with time delays. Firstly, a new double integral inequality is proposed. Then, it is used to derive a new delay-dependent stability criterion in terms of linear matrix inequalities (LMIs). Two numerical examples are given to demonstrate the effectiveness and merits of the present result.

Functional observer design for retarded system with interval time-varying delays

ABSTRACTFunctional observers are the key solution to numerous practical estimation problems, wherein full-order observers cannot be applied. This paper studies the novel problem of minimum-order functional observer design for time-delay systems with interval time-varying state delays. Moreover, unlike the majority of the existing papers on this topic, which consider either small or unknown delay rates, the delay derivative is assumed to possess an upper bound not limited to be less than one. A new augmented Lyapunov–Krasovskii functional with triple integral terms is employed to derive less conservative delay-dependent sufficient conditions for the stability of the closed-loop observer dynamics, which are expressed in terms of linear matrix inequalities. In addition, contemporary techniques such as Wirtinger-based single- and double-integral inequalities, delay splitting scheme, reciprocally convex approach and convex combination technique, along with the descriptor transformation, are adopted in constructing the new stability criterion that is exploited for obtaining the observer parameters. A genetic-algorithm-based searching schema is proposed to optimally tune a number of weighting parameters in the observer design procedure. Numerical examples and simulation results are demonstrated to confirm the effectiveness and the superiority of the proposed observer design algorithm.

Wirtinger-based multiple integral inequality approach to synchronization of stochastic neural networks

This paper investigates the synchronization problem for a class of chaotic neural networks with discrete and unbounded distributed time delays under stochastic perturbations. Firstly, based on the Wirtinger-based double integral inequality, two novel inequalities are proposed, which are multiple integral forms of the Wirtinger-based integral inequality. Next, by applying the Jensen-type integral inequality for stochastic case and combining the Jensen integral inequality with the reciprocally convex combination approach, a delay-dependent criterion is developed to achieve the synchronization for the stochastic chaotic neural networks in the sense of mean square. In the case of no stochastic perturbations, by applying the reciprocally convex combination approach for high order case and a free-matrix-based inequality, novel delay-dependent conditions are established to achieve the synchronization for the chaotic neural networks. All the results are based on dividing the bounding of activation function into two subintervals with equal length. Finally, two numerical examples are provided to demonstrate the effectiveness of the theoretical results.

Sampled‐data reliable stabilization of T‐S fuzzy systems and its application

In this article, based on sampled‐data approach, a new robust state feedback reliable controller design for a class of Takagi–Sugeno fuzzy systems is presented. Different from the existing fault models for reliable controller, a novel generalized actuator fault model is proposed. In particular, the implemented fault model consists of both linear and nonlinear components. Consequently, by employing input‐delay approach, the sampled‐data system is equivalently transformed into a continuous‐time system with a variable time delay. The main objective is to design a suitable reliable sampled‐data state feedback controller guaranteeing the asymptotic stability of the resulting closed‐loop fuzzy system. For this purpose, using Lyapunov stability theory together with Wirtinger‐based double integral inequality, some new delay‐dependent stabilization conditions in terms of linear matrix inequalities are established to determine the underlying system's stability and to achieve the desired control performance. Finally, to show the advantages and effectiveness of the developed control method, numerical simulations are carried out on two practical models. © 2016 Wiley Periodicals, Inc. Complexity 21: 518–529, 2016

Robust finite time stabilization analysis for uncertain neural networks with leakage delay and probabilistic time-varying delays

This paper investigates the problem of robust finite time stabilization for a uncertain neural networks with leakage delay and probabilistic time-varying delays. By introducing a stochastic variable which satisfies Bernoulli distribution, the information of probabilistic time-varying delay is equivalently transformed into the deterministic time-varying delay with stochastic parameters. The main objective of this paper is to design a memoryless state feedback control such that the resulting proposed system is robustly finite time stable with admissible uncertainties. Based on a suitable Lyapunov–Krasovskii functional, model transformation technique and Wirtinger-based double integral inequality, the general framework is obtained in terms of linear matrix inequalities to determine the finite time stability and to achieve the control design. Finally, three numerical examples are presented to validate the effectiveness and less conservatism of the proposed method.

New delay-interval-dependent stability criteria for switched Hopfield neural networks of neutral type with successive time-varying delay components.

This paper deals with the problem of delay-interval-dependent stability criteria for switched Hopfield neural networks of neutral type with successive time-varying delay components. A novel Lyapunov-Krasovskii (L-K) functionals with triple integral terms which involves more information on the state vectors of the neural networks and upper bound of the successive time-varying delays is constructed. By using the famous Jensen's inequality, Wirtinger double integral inequality, introducing of some zero equations and using the reciprocal convex combination technique and Finsler's lemma, a novel delay-interval dependent stability criterion is derived in terms of linear matrix inequalities, which can be efficiently solved via standard numerical software. Moreover, it is also assumed that the lower bound of the successive leakage and discrete time-varying delays is not restricted to be zero. In addition, the obtained condition shows potential advantages over the existing ones since no useful term is ignored throughout the estimate of upper bound of the derivative of L-K functional. Using several examples, it is shown that the proposed stabilization theorem is asymptotically stable. Finally, illustrative examples are presented to demonstrate the effectiveness and usefulness of the proposed approach with a four-tank benchmark real-world problem.

On the Hadamard’s type inequalities for co-ordinated convex functions via fractional integrals

In this paper, we establish two identities for functions of two variables and apply them to give new Hermite–Hadamard type fractional integral inequalities for double fractional integrals involving functions whose derivatives are bounded or co-ordinates convex function on Δ≔[a,b]×[c,d] in R2 with a<b,c<d.

Stability of Markovian Jump Generalized Neural Networks With Interval Time-Varying Delays

This paper examines the problem of asymptotic stability for Markovian jump generalized neural networks with interval time-varying delays. Markovian jump parameters are modeled as a continuous-time and finite-state Markov chain. By constructing a suitable Lyapunov-Krasovskii functional (LKF) and using the linear matrix inequality (LMI) formulation, new delay-dependent stability conditions are established to ascertain the mean-square asymptotic stability result of the equilibrium point. The reciprocally convex combination technique, Jensen's inequality, and the Wirtinger-based double integral inequality are used to handle single and double integral terms in the time derivative of the LKF. The developed results are represented by the LMI. The effectiveness and advantages of the new design method are explained using five numerical examples.