Some Novel Estimates of Hermite–Hadamard and Jensen Type Inequalities for (h1,h2)-Convex Functions Pertaining to Total Order Relation

There are different types of order relations that are associated with interval analysis for determining integral inequalities. The purpose of this paper is to connect the inequalities terms to total order relations, often called (CR)-order. In contrast to classical interval-order relations, total order relations are quite different and novel in the literature and are calculated as ω=⟨ωc,ωr⟩=⟨ω¯¯+ω−−2,ω¯¯−ω−−2⟩. A major benefit of total order relations is that they produce more efficient results than other order relations. This study introduces the notion of CR-(h1,h2)-convex function using total order relations. Center and Radius order relations are a powerful tool for studying inequalities based on their properties and widespread application. Using this novel notion, we first developed some variants of Hermite–Hadamard inequality and then constructed Jensen inequality. Based on the results, this new concept is extremely useful in connection with a variety of inequalities. There are many new and well-known convex functions unified by this type of convexity. These results will stimulate further research on inequalities for fractional interval-valued functions and fuzzy interval-valued functions, as well as the optimization problems associated with them. For the purpose of verifying our main findings, we provide some nontrivial examples

An equivalent condition to the Jensen inequality for the generalized Sugeno integral.

For the classical Jensen inequality of convex functions, i.e., [Formula: see text] an equivalent condition is proved in the framework of the generalized Sugeno integral. Also, the necessary and sufficient conditions for the validity of the discrete form of the Jensen inequality for the generalized Sugeno integral are given

A review of convex approaches for control, observation and safety of linear parameter varying and Takagi-Sugeno systems

This paper provides a review about the concept of convex systems based on Takagi-Sugeno, linear parameter varying (LPV) and quasi-LPV modeling. These paradigms are capable of hiding the nonlinearities by means of an equivalent description which uses a set of linear models interpolated by appropriately defined weighing functions. Convex systems have become very popular since they allow applying extended linear techniques based on linear matrix inequalities (LMIs) to complex nonlinear systems. This survey aims at providing the reader with a significant overview of the existing LMI-based techniques for convex systems in the fields of control, observation and safety. Firstly, a detailed review of stability, feedback, tracking and model predictive control (MPC) convex controllers is considered. Secondly, the problem of state estimation is addressed through the design of proportional, proportional-integral, unknown input and descriptor observers. Finally, safety of convex systems is discussed by describing popular techniques for fault diagnosis and fault tolerant control (FTC).Peer ReviewedPostprint (published version