Discrete Jensen Inequality Research Articles

Necessary and sufficient conditions for discrete inequalities of Jensen–Steffensen type with applications

AbstractIn this paper we give a necessary and sufficient condition for the discrete Jensen inequality to be satisfied for real (not necessarily nonnegative) weights. The result generalizes and completes the classical Jensen–Steffensen inequality. The validity of the strict inequality is studied. As applications, we first give the form of our result for strongly convex functions, then we study discrete quasi-arithmetic means with real (not necessarily nonnegative) weights, and finally we refine the inequality obtained.

External Jensen-type operator inequalities via superquadraticity

In this paper we establish several Jensen-type operator inequalities for a class of superquadratic functions and self-adjoint operators. Our results are given in the so-called external form. As an application, we give improvements of the H?lder-McCarthy inequality and the classical discrete and integral Jensen inequality in the corresponding external forms. In addition, the established Jensen-type inequalities are compared with the previously known results and we show that our results provide more accurate estimates in some general settings.

Discrete Legendre polynomials-based inequality for stability of time-varying delayed systems

This paper proposes Discrete Legendre Polynomial(DLP)-based inequality by solving the best weighted approximation of a given time series. The proposed inequality could significantly reduce the conservativeness in stability analysis of systems with constant or interval time-varying delays. Also former well-known integral inequities, such as discrete Jensen inequality, discrete Wirtinger-based inequality, are both included in the proposed DLP-based inequality as special cases with lower-order approximation. Stability criterion with less conservatism is then developed for both constant and time-varying delayed systems. Several numerical examples are given to demonstrate the effectiveness and benefit of the proposed method.

Delay-Dependent Stability and $$H_{\infty }$$ H ∞ Control for 2-D Markovian Jump Delay Systems with Missing Measurements and Sensor Nonlinearities

This paper investigates the problem of stability and $${H}_\infty $$H? control for 2-D time-delayed Markovian jump systems with missing measurements and sensor nonlinearities. The measured signals transmitted between the system and the controller are supposed to be imperfect, which involves missing measurements and sensor nonlinearities. The data missing is described by a random variable that obeys the Bernoulli binary distribution, and the sensor nonlinearities are assumed to satisfy the sector conditions. The problem addressed is the design of an output feedback controller such that the resulting closed-loop system is mean-square asymptotically stable and has a prescribed $${H}_\infty $$H? performance level. By using the Lyapunov method and the discrete Jensen inequality, stability criteria and $${H}_\infty $$H? performance level are established, and then, the controller design problem is cast into optimization problems via two methods. Finally, a numerical example is exploited to illustrate the effectiveness of the proposed design method.

Cyclic Refinements of the Different Versions of Operator Jensen's Inequality

Refinements of the operator Jensen's inequality for convex and operator convex functions are given by using cyclic refinements of the discrete Jensen's inequality. Similar refinements are fairly rare in the literature. Some applications of the results to norm inequalities, the Holder McCarthy inequality and generalized weighted power means for operators are presented.

L∞-gain performance analysis for two-dimensional Roesser systems with persistent bounded disturbance and saturation nonlinearity

The l∞-gain approach has been an essential tool in one-dimensional system theory. However, limited results have been presented in the literature for the two-dimensional (2-D) l∞-gain approach. This paper investigates the l∞-gain performance for 2-D systems in the Roesser model with persistent bounded disturbance input and saturation nonlinearity. A linear matrix inequality (LMI)-based condition is established to reduce the effect of persistent bounded disturbance input on 2-D systems within a given disturbance attenuation level based on the discrete Jensen inequality, lower bounds lemma, and diagonally dominant matrices. We apply the obtained results to the l∞-gain performance analysis for 2-D digital filters with saturation arithmetic.

Less conservative stability condition for uncertain discrete-time recurrent neural networks with time-varying delays

This paper is concerned with the stability analysis problem for uncertain stochastic discrete-time recurrent neural networks with time-varying delay. By using linear matrix inequality method and discrete Jensen inequality, a new Lyapunov–Krasovskii function is established to derive sufficient condition for globally asymptotical stability in mean square of the recurrent neural networks with stochastic disturbance. As an extension, we further consider the stability analysis problem for the same class of neural networks but with uncertainty. It is shown that the newly obtained result is less conservative than the existing ones when the described system is without disturbance and uncertainty. Meanwhile, the computational complexity is reduced since less variable is involved. Two numerical examples are presented to illustrate the effectiveness and the benefits of the proposed method.

Discrete Wirtinger-based inequality and its application

In this paper, we derive a new inequality, which encompasses the discrete Jensen inequality. The new inequality is applied to analyze stability of linear discrete systems with an interval time-varying delay and a less conservative stability condition is obtained. Two numerical examples are given to show the effectiveness of the obtained stability condition.

ESTIMATED STATE FEEDBACK FUZZY CONTROL FOR PASSIVE DISCRETE TIME-DELAY MULTIPLICATIVE NOISED PENDULUM SYSTEMS

This paper develops a fuzzy controller design method via estimated state feedback scheme for discrete-time nonlinear stochastic time-delay systems which are presented by Takagi-Sugeno fuzzy model with multiplicative noises. The proposed design method is accomplished by the concept of parallel distributed compensation. Using the Lyapunov-Krasovskii function and passivity theory, the stability conditions are derived to guarantee the stability and attenuation performance. An algorithm is proposed in this paper to solve the derived stability conditions that belong to non-strict linear matrix inequality problems. Moreover, the discrete Jensen inequality and free-weighting matrix technique are employed to decrease the conservatism of the proposed design method. Finally, the control of a nonlinear time-delay pendulum system is provided to illustrate the effectiveness and usefulness of the proposed fuzzy control method.

Infinite refinements of the discrete Jensen's inequality defined by recursion

Infinite refinements of the discrete Jensen's inequality defined by recursion

Delay-dependent [formula omitted] filtering for discrete-time fuzzy stochastic systems with mixed delays and sector-bounded nonlinearity

The delay-dependent H∞ filter problem for discrete-time Takagi–Sugeno (T–S) fuzzy stochastic systems with mixed delays and sector-bounded nonlinearity is considered in this paper. The mixed delays consist of both discrete and distributed delays, and the nonlinearity under consideration is subject to sector-bounded condition. Both the the fuzzy-rule-dependent and fuzzy-rule-independent filters are considered. Based on discrete Jensen inequality and the Lyapunov–Krasovskii functional approach combined with the S-procedure, sufficient conditions for the existence of admissible filters are established in terms of linear matrix inequalities (LMIs), which ensure the asymptotical mean-square stability as well as a prescribed H∞ performance. Finally, numerical examples are given to demonstrate the effectiveness and less conservatism of the proposed approach.

A Class of Nonlinear Weakly Singularity Wendroff Type Integral Inequality with Two Variables and Application

In this paper, we establish a nonlinear weakly singularity Wendroff type integral inequality with two variables, which generalizes the unknown function with a variable to composite function of nonlinear function with unknown function with two variables. Under certain conditions, the estimation of unknown function is given by the technique of amplification, variable substitutions, integration and differentiation, discrete Jensen inequality.

Estimation of the Unknown Function of Integral Inequality with Power Function and its Application

In this paper, we establish a new weakly singularity integral inequality with two variables, which generalize the unknown function with a variable to composite function of power function with unknown function with two variables. Under certain conditions, the estimation of unknown function is given by technique of change of variable, amplification method, integration and differentiation, discrete Jensen inequality and inequality.

A Class of Nonlinear Singular Difference Inequality in Engineering

In this paper, we discuss a class of new nonlinear weakly singular difference inequality. Using change of variable, discrete Jensen inequality, amplification method, the mean-value theorem for integrals and Gamma function, explicit bounds for the unknown functions in the inequality is given clearly. The derived results can be applied in the study of fractional difference equations in Engineering.

A Class of Singular Volterra-Fredholm Type Difference Inequality in Engineering

In this paper, we discuss a class of new weakly singular Volterra-Fredholm difference inequality, which is solved using change of variable, discrete Jensen inequality, Beta function, the mean-value theorem for integrals and amplification method, and explicit bounds for the unknown functions is given clearly. The derived results can be applied in the study of fractional difference equations in engineering.

A Class of Volterra-Fredholm Difference Inequality with Weakly Singularity in Engineering

In this paper, we discuss a class of new Volterra-Fredholm weakly singular difference inequality. The explicit bounds for the unknown functions are given clearly by discrete Jensen inequality, Cauchy-Schwarz inequality, Gamma function, change of variable, the mean-value theorem for integrals and amplification method. The derived results can be applied in the study of fractional difference equations in engineering.

Estimation of Unknown Function in a Class of Singular Difference Inequality in Engineering

In this paper, we discuss a class of new weakly singular difference inequality, which is solved using change of variable, discrete Jensen inequality, Beta function, the mean-value theorem for integrals and amplification method, and explicit bounds for the unknown functions is given clearly. The derived results can be applied in the study of fractional difference equations in Engineering.

Weighted form of a recent refinement of the discrete Jensen's inequality

Recently, Xiao, Srivastava and Zhang (see [10]) have introduced a new refinement of the discrete Jensen’s inequality for mid-convex functions. We give and discuss the weighted form of their results. This leads to some new inequlities and limit formulas. We illustrate the scope of the results by applying them to introduce and study some new quasi-arithmetic means. Mathematics subject classification (2010): 26D07, 26A51.

A Class of Iterative Nonlinear Difference Inequality with Weakly Singularity

We discuss a class of new nonlinear weakly singular difference inequality, which is solved by change of variable, discrete Hölder inequality, discrete Jensen inequality, the mean-value theorem for integrals and amplification method, and Gamma function. Explicit bound for the unknown function is given clearly. Moreover, an example is presented to show the usefulness of our results.

Imperfect premise matching based fuzzy control with passive constraints for discrete time-delay multiplicative noised stochastic nonlinear systems

This paper investigates a fuzzy controller design method for discrete-time nonlinear stochastic time-delay systems which are presented by the Takagi-Sugeno (T-S) fuzzy model with multiplicative noises. Utilizing the proposed design method, the fuzzy controller can be carried out via not only state feedback scheme but also output feedback scheme. Both of them are accomplished by the concept of imperfect premise matching (IPM). For discussing the stabilization problem, the Lyapunov-Krasovskii function and passivity theory are applied to derive the sufficient conditions. Moreover, the discrete Jensen inequality is employed to decrease the conservatism of the proposed method. Finally, a numerical example for the control of a nonlinear time-delay pendulum system is provided to show the effectiveness and usefulness of the proposed design method.