Search

Abstract 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.

A counterpart of the Ohlin theorem for convex set-valued maps is proved. An application of this result to obtain some inclusions related to convex set-valued maps in an alternative unified way is presented. In particular counterparts of the Jensen integral and discrete inequalities, the converse Jensen inequality and the Hermite–Hadamard inequalities are obtained. [ABSTRACT FROM AUTHOR]

Since 2004 more than 80 articles were published related to superquadracity. Here we put together some of these results, in particular those related to Jensen's inequality in one and in several variables, to Jensen-Steffensen inequalities, to Hardy's inequalities and to Hermite-Hadamard and Fejer inequalities. These inequalities are analogs of inequalities satisfied by convex functions and in those cases that the superquadratic functions are non-negative we get refinements of inequalities satisfied by convex functions. [ABSTRACT FROM AUTHOR]

The squared modulus of every real-valued on R function f from the Laguerre–Pólya class L – P obeys a MacLaurin-type series representation | f ( x + i y ) | 2 = ∑ k = 0 ∞ L k ( f ; x ) y 2 k , x , y ∈ R . If f is a polynomial with only real roots, then the sum becomes finite. The coefficients { L k } are representable as non-linear differential operators acting on f , and by a classical result of Jensen L k ( f ; x ) ≥ 0 for f ∈ L – P and x ∈ R . A conjecture of M. Patrick from 1971 states that for f = P n ( α , β ) , the n -th Jacobi polynomial, with α ≥ β > − 1 , the functions L k ( f ; x ) , 1 ≤ k ≤ n − 1 , attain their maxima in [ 0 , 1 ] at x = 1 . The aim of this paper is to validate Patrick's conjecture. Moreover, we prove a refined version of this conjecture, showing that { L k ( f ; x ) } k = 1 n − 1 are strictly monotonically increasing functions on the positive semi-axis. Towards our proof of Patrick's conjecture we extend the Sonin–Pólya majorization approach to all coefficient functions { L k ( f ; x ) } k = 0 n , f = P n ( α , β ) . [ABSTRACT FROM AUTHOR]

This study deals with the problem of delay‐dependent dissipative filtering for genetic regulatory networks (GRNs) with norm‐bounded parameter uncertainties and time‐varying delays. It is assumed that the non‐linear function describing the feedback regulation satisfies the sector‐bounded condition. Improved delay‐dependent stochastic stability and filter design method for stochastic GRNs are obtained by applying the delay fractioning technique, Jensen inequalities and introducing some proper slack matrix variables. The conservatism caused by either model transformation or bounding techniques is reduced. Sufficient conditions of the robust stability and filtering problems are proposed, respectively. The filter, which can guarantee the resulting error system to be asymptotically stable in the mean‐square sense and satisfy a prescribed performance level for all delays no larger than a given upper bound, are constructed. A numerical example is provided to demonstrate effectiveness of the proposed results in this study. [ABSTRACT FROM AUTHOR]

It is well-known that the squared modulus of every function f from the Laguerre-Polya class $${\mathcal{L}-\mathcal{P}}$$ of entire functions obeys a MacLaurin series representation, which reduces to a finite sum when f is a polynomial having only real zeros. The coefficients { L} are representable as non-linear differential operators acting on f, and by a classical result of Jensen L( f; x) ≥ 0 for $${f\in \mathcal{L}-\mathcal{P}}$$ and $${x\in \mathbb{R}}$$ . Here, we prove a conjecture formulated by the first-named author in 2005, which states that for $${f=P_n^{(\lambda)} }$$ , the n-th Gegenbauer polynomial, the functions $${\{L_k(f;x)\}_{k=1}^{n}}$$ are monotone decreasing on the negative semi-axis and monotone increasing on the positive semi-axis. This result pertains to certain polynomial inequalities in the spirit of the celebrated refinement of Markov's inequality, found by R. J. Duffin and A. C. Schaeffer in 1941. [ABSTRACT FROM AUTHOR]