Jensen Integral Inequality Research Articles

Delay-Dependent Globally Exponential Stability Criteria for Static Neural Networks: An LMI Approach

The problem of globally exponential stability of static neural networks is investigated. Based on the Lyapunov-Krasovskii functional approach, the free-weighting matrix method, and the Jensen integral inequality, new delay-dependent stability criteria of the unique equilibrium of static neural networks with time-varying delays are presented in terms of linear matrix inequalities (LMIs). The stability criteria can easily be checked by using recently developed algorithms in solving LMIs. A numerical example is given to illustrate the effectiveness and less conservativeness of our proposed method.

New LMT-based delay-dependent criterion for global asymptotic stability of cellular neural networks

The problem of global asymptotic stability analysis is studied for a class of cellular neural networks with time-varying delay. By defining a Lyapunov–Krasovskii functional, a new delay-dependent stability condition is derived in terms of linear matrix inequalities. The obtained criterion is less conservative than some previous literature because free-weighting matrix method and the Jensen integral inequality are considered. Three illustrative examples are given to demonstrate the effectiveness of the proposed results.

Jensen integral inequality approach to stability analysis of continuous-time systems with time-varying delay

The problem of stability analysis for continuous-time systems with time-varying delay is studied. By defining novel Lyapunov functionals and using the Jenson integral inequality, new delay-dependent stability conditions are obtained in terms of linear matrix inequalities. Unlike previous methods, the upper bound of the delay derivative is taken into consideration, and this upper bound is allowed to be greater than or equal to 1. It is proved that the newly proposed criteria may introduce less conservatism than some existing ones. Meanwhile, the computational complexity of the presented stability criteria is reduced greatly since fewer decision variables are involved. Numerical examples are given to illustrate the effectiveness of the proposed methods.

Interpolations of Jensen's inequality

Weighted and unweighted interpolations of general order are given for Jensen's integral inequality. Various upper-bound estimates are made for the differences between the interpolates and some convergence results erived. The results generalise and subsume a body of earlier work and employ streamlined proofs.

An Integral Jensen Inequality For Convex Multifunctions

We prove the following multivalued version of the Jensen integral inequality. Let X, Y be Banach spaces and D ⊂ X an open and convex set. If F: D ↦ cl(Y) is a continuous convex function, then for each normalized measure space (Ω, S, μ), and for all μ-integrable functions ϕ : Ω ↦ D such that convϕ(Ω) ⊂ D, % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A! $$\int_{\Omega}(F\ o\ \phi)d\mu \subset F\Bigg(\int_{\Omega}\phi d\mu\Bigg).$$

A continuous basis for Orlicz spaces

1. In 1910 Haar created the orthonormal system that bears his name in order to show that there are ON systems with respect to which the Fourier series of each continuous function converges uniformly to the function. The Haar functions are themselves discontinuous, however, and this led Franklin to construct a continuous ON set that plays the same role. Now the Haar functions comprise a Schauder basis for each of the spaces LP [0, 1], p _ 1, as Schauder himself has shown [5]. Indeed, this system serves as a Schauder basis for each of the separable Orlicz spaces associated with the unit interval, see [2], [4]. Thus, it is natural to inquire whether the Franklin system also serves as a basis for these spaces. In the present article this question is answered affirmatively. The proof is elementary, requiring only the judicious application of the uniform boundedness principle and the Jensen integral inequality. For the spaces LP, p _ 1, this result has been noted recently by Ciesielski [1]. Nevertheless, even for this special case, the demonstration given below may be of interest by virtue of its utter simplicity.