Boundedness Properties Research Articles

Robust nonlinear observer-based controller design for one-sided Lipschitz switched systems with time-varying delays

The core objective of this paper is to investigate the problem of robust nonlinear observer-based controller design for one-sided Lipschitz (OLS) switched systems. This work considers nonlinear switched systems with norm-bounded uncertainties, time-varying delays, and external disturbances. By selecting appropriate Lyapunov-Krasovskii functions and applying the OSL condition together with the quadratic inner boundedness property, we drive certain necessary and sufficient conditions for the observer-based control of the underlying system that assure the exponential convergence of the system states, the observer states, and the error states to zero, in respect of linear matrix inequalities via average dwell-time approach. The devised scheme offers necessary robustness against L2-norm bounded perturbations. In the end, the proposed scheme has been validated a suitable practical example.

A solving method for two-dimensional homogeneous system of fuzzy fractional differential equations

<abstract> <p>The purpose of this study is to extend and determine the analytical solution of a two-dimensional homogeneous system of fuzzy linear fractional differential equations with the Caputo derivative of two independent fractional orders. We extract two possible solutions to the coupled system under the definition of strongly generalized $ H $-differentiability, uncertain initial conditions and fuzzy constraint coefficients. These potential solutions are determined using the fuzzy Laplace transform. Furthermore, we extend the concept of fuzzy fractional calculus in terms of the Mittag-Leffler function involving triple series. In addition, several important concepts, facts, and relationships are derived and proved as property of boundedness. Finally, to grasp the considered approach, we solve a mathematical model of the diffusion process using proposed techniques to visualize and support theoretical results.</p></abstract>

Information Fusion Model of Group Decision Making Based on a Combinatorial Ordered Weighted Average Operator

Aiming at the problem that the existing aggregation operators only consider the weight of the data itself or the weight of the data position when aggregating data, a group decision information fusion model based on the combinatorial ordered weighted average operator is proposed, which considers the incompleteness, uncertainty and plurality of the decision information, and effectively improves the accuracy of group consensus. In this paper, an interval intuitionistic fuzzy combinatorically weighted average operator is proposed, which embodies the weight of the set data itself and the weight of data position, and can adjust the importance of the two weights in the operator through parameters, and prove that the operator has the properties of boundedness, monotonicity, idempotency and permutation invariance. Secondly, a position weight solving model of interval intuitionistic fuzzy combinatorial ordered weighted average operator is established, and the position weight is solved with the help of cross-entropy and Orness measures. Finally, a group decision information fusion process based on IVIFCOWA operators is designed, and the decision maker weight information is obtained according to the decision matrix, and the comprehensive decision matrix is obtained by fusion of IVIFCOWA operators, and then the mean similarity is obtained. Through specific case analysis, it is proved that the proposed operator can make full use of the decision data, and can better reach group consensus after data fusion than the existing operator.

Robust Adaptive Prescribed Performance Control for Unknown Nonlinear Systems With Input Amplitude and Rate Constraints

A novel approximation-free prescribed performance control scheme for unknown nonlinear systems with amplitude and rate saturation on the control input signal is designed in this letter. The proposed control strategy employs nested smooth saturation functions and introduces a reconciling relaxation of the performance constraints based on the actuator limitations. The straightforward gain selection along with the low complexity of the control scheme, simplifies the practical implementation of our algorithm. The adoption of the adaptive prescribed performance control technique ensures the desired trade-off between input and output constraints for any input-to-state stable (ISS) system. Inevitably, for generic systems, the boundedness properties are guaranteed only locally; thus, we provide a sufficient boundedness condition for all closed-loop signals. Finally, an illustrative simulation study is conducted to demonstrate the efficacy of the proposed approach.

Prioritization of Strategies of Digital Transformation of Supply Chain Employing Bipolar Complex Fuzzy Linguistic Aggregation Operators

Digitalization has addressed all parts of organizations together with supply chain management. Innovations like installed RFID, GPS, and sensors have assisted organizations with changing their current conventional supply chain framework into more spry, adaptable, open, and cooperative computerized models. A lot of organizations want to get the advantages of digitalization, particularly in the supply chain But here, the biggest problem, that the organizations are facing is the ambiguities, i.e. how to select and evaluate the best strategy for the supply chain digital transformation (SCDT), on which criteria or attribute they can assess the strategies of SCDT, etc. Multi-attribute decision-making (MADM) procedure is the finest technique to evaluate and find out the finest strategy of the SCDT. To handle these ambiguities and for making the MADM technique, this study first establishes the concept of bipolar complex fuzzy linguistic sets (BCFLSs) and its essential properties. Furthermore, we scrutinize average and geometric AOs for BCFLS such as bipolar complex fuzzy linguistic weighted averaging (BCFLWA) and bipolar complex fuzzy linguistic weighted geometric (BCFLWG) operators along with their idempotency, boundedness, and monotonicity properties. After that, to exhibit the usefulness of our established notion and AOs in real life and the selection and evaluation of the strategy of digital transformation of the supply chain, we interpret a DM technique and explore a numerical example. To exhibit the supremacy of the defined BCFLS and defined AOs we compare them with certain prevailing conceptions.

An n-dimensional pseudo-differential operator involving linear canonical transform and some applications in quantum mechanics

In this work, an n-dimensional pseudo-differential operator involving the n-dimensional linear canonical transform associated with the symbol ?(x1,..., xn; y1,..., yn) ? C?(Rn ? Rn) is defined. We have introduced various properties of the n-dimensional pseudo-differential operator on the Schwartz space using linear canonical transform. It has been shown that the product of two n-dimensional pseudodifferential operators is an n-dimensional pseudo-differential operator. Further, we have investigated formal adjoint operators with a symbol ? ? Sm using the n-dimensional linear canonical transform, and the Lp(Rn) boundedness property of the n-dimensional pseudo-differential operator is provided. Furthermore, some applications of the n-dimensional linear canonical transform are given to solve generalized partial differential equations and their particular cases that reduce to well-known n-dimensional time-dependent Schr?dinger-type-I/Schr?dinger-type-II/Schr?dinger equations in quantum mechanics for one particle with a constant potential.

Safe and Robust Observer-Controller Synthesis Using Control Barrier Functions

This paper addresses the synthesis of safety-critical controllers using estimate feedback. We propose an observer-controller interconnection to ensure that the nonlinear system remains safe despite bounded disturbances on the system dynamics and measurements that correspond to partial state information. The co-design of observers and controllers is critical, since even in undisturbed cases, observers and controllers designed independently may not render the system safe. We propose two approaches to synthesize observer-controller interconnections. The first approach utilizes Input-to-State Stable observers, and the second uses Bounded Error observers. Using these stability and boundedness properties of the observation error, we construct novel Control Barrier Functions that impose inequality constraints on the control inputs which, when satisfied, certifies safety. We propose quadratic program-based controllers to satisfy these constraints, and prove Lipschitz continuity of the derived controllers. Simulations and experiments on a quadrotor demonstrate the efficacy of the proposed methods.

Accurate trajectory tracking control with adaptive neural networks for omnidirectional mobile robots subject to unmodeled dynamics

Omnidirectional mobile robots have gained a lot of attention in recent years due to their maneuverability capabilities. However, ensuring accurate trajectory tracking with this class of robots is still challenging control system designers. In this work, a novel intelligent controller is introduced for accurate trajectory tracking of omnidirectional robots subject to unstructured uncertainties. An adaptive neural network is adopted within a Lyapunov-based nonlinear control scheme to deal with frictional forces and other unmodeled dynamics or external disturbances that may occur. Online learning, rather than supervised offline training, is employed to allow the robot to learn on its own how to compensate for uncertainties and disturbances by interacting with the environment. The adoption of a combined error signal as the single input in the neural network significantly reduces the computational complexity of the disturbance compensation scheme and enables the resulting intelligent controller to be implemented in the embedded hardware of mobile robots. The boundedness and convergence properties of the proposed control scheme are proved by means of a Lyapunov-like stability analysis. The effectiveness of the proposed intelligent controller is numerically evaluated and experimentally validated using an omnidirectional mobile robot. The comparative analyses of the obtained results show that the adoption of an intelligent compensation scheme based on adaptive neural networks allows reductions of more than 95% in the tracking error, thus guaranteeing an accurate tracking and confirming the great superiority of the proposed control strategy.

On the characterization properties of certain hypergeometric functions in the open unit disk

Our purpose in the present investigation is to study certain geometric properties such as the close-to-convexity, convexity, and starlikeness of the hypergeometric function z{}_{1}F_{2}(a;b,c;z) in the open unit disk. The usefulness of the main results for some familiar special functions like the modified Sturve function, the modified Lommel function, the modified Bessel function, and the {}_{0}F_{1}(-;c;z) function are also mentioned. We further consider a boundedness property of the function _{1}F_{2}(a;b,c;z) in the Hardy space of analytic functions. Several corollaries and special cases of the main results are also pointed out.

Novel Complex Pythagorean Fuzzy Sets under Aczel–Alsina Operators and Their Application in Multi-Attribute Decision Making

Aggregation operators (AOs) are utilized to overcome the influence of uncertain and vague information in different fuzzy environments. A multi-attribute decision-making (MADM) technique plays a vital role in several fields of different environments such as networking analysis, risk assessment, cognitive science, recommender systems, signal processing, and many more domains in ambiguous circumstances. In this article, we elaborated the notion of Aczel–Alsina t-norm (TNM) and t-conorm (TCNM) under the system of complex Pythagorean fuzzy (CPyF) sets (CPyFSs). Some basic operational laws of Aczel–Alsina TNM and TCNM are established including Aczel–Alsina sum, product, scalar multiplication, and power operations based on CPyFSs. We established several AOs of CPyFSs such as CPyF Aczel–Alsina weighted average (CPyFAAWA), and CPyF Aczel–Alsina weighted geometric (CPyFAAWG) operators. The proposed CPyFAAWA and CPyFAAWG operators are symmetric in nature and satisfy the properties of idempotency, monotonicity, boundedness and commutativity. To solve an MADM technique, we established an illustrative example to select a suitable candidate for a vacant post in a multinational company. To see the advantages of our proposed AOs, we compared the results of existing AOs with the results of newly established AOs.

Certain averaging operators on Triebel-Lizorkin spaces

In this article, we study the boundedness properties of the averaging operator Stγ on Triebel-Lizorkin spaces dot F_{p,q}^alpha left( {{mathbb{R}^n}} right) for various p,q. As an application, we obtain the norm convergence rate for Stγ (f) on Triebel-Lizorkin spaces and the relation between the smoothness imposed on functions and the rate of norm convergence of Stγ is given.

Modeling the effect of contaminated object, vaccination and treatment on the diseases epidemiology

In this paper, the effect of contaminated object on an SIRS Model with vaccination and hospitalization compartments is modeled. Positivity and boundedness properties of the solutions of model are proved, basic reproduction number of the model is founded through criteria which make the eigen values of the Jacobean matrix at the disease free equilibrium point, negative. Globally stability analysis of the disease free equilibrium point is proved when the basic reproduction number is less than unity and the vaccination efficacy approaches unity. The existence, uniqueness of the endemic equilibrium point are investigated when the basic reproduction number greater than unity. Parameter values regarding to spreading covid-19 in Kurdistan region are estimated. Finally, sensitivity analysis of the reproduction number is carried out.

A Multi-Attribute Decision-Making Approach for the Analysis of Vendor Management Using Novel Complex Picture Fuzzy Hamy Mean Operators

Vendor management systems (VMSs) are web-based software packages that can be used to manage businesses. The performance of the VMSs can be assessed using multi-attribute decision-making (MADM) techniques under uncertain situations. This article aims to analyze and assess the performance of VMSs using MADM techniques, especially when the uncertainty is of complex nature. To achieve the goals, we aim to explore Hany mean (HM) operators in the environment of complex picture fuzzy (CPF) sets (CPFSs). We introduce CPF Hamy mean (CPFHM) and CPF weighted HM (CPFWHM) operators. Moreover, the reliability of the newly proposed HM operators is examined by taking into account the properties of idempotency, monotonicity, and boundedness. A case study of VMS is briefly discussed, and a comprehensive numerical example is carried out to assess VMSs using the MADM technique based on CPFHM operators. The sensitivity analysis and comprehensive comparative analysis of the proposed work are discussed to point out the significance of the newly established results.

Discrete Hölder spaces and their characterization via semigroups associated with the discrete Laplacian and kernel estimates

In this paper, we characterize the discrete Hölder spaces by means of the heat and Poisson semigroups associated with the discrete Laplacian. These characterizations allow us to get regularity properties of fractional powers of the discrete Laplacian and the Bessel potentials along these spaces and also in the discrete Zygmund spaces in a more direct way than using the pointwise definition of the spaces. To obtain our results, it has been crucial to get boundedness properties of the heat and Poisson kernels and their derivatives in both space and time variables. We believe that these estimates are also of independent interest.

Properties of Hadamard Fractional Integral and Its Application

We begin by introducing some function spaces Lcp(R+),Xcp(J) made up of integrable functions with exponent or power weights defined on infinite intervals, and then we investigate the properties of Mellin convolution operators mapping on these spaces, next, we derive some new boundedness and continuity properties of Hadamard integral operators mapping on Xcp(J) and Xp(J). Based on this, we investigate a class of boundary value problems for Hadamard fractional differential equations with the integral boundary conditions and the disturbance parameters, and obtain uniqueness results for positive solutions to the boundary value problem under some weaker conditions.

A study on the Heronian mean operators for managing complex picture fuzzy uncertain linguistic settings and their application in decision making

The major influence of this manuscript is to diagnose a valuable and considerable technique of Complex Picture Fuzzy Uncertain Linguistic setting and described its useful and valuable operational laws. The theory of Complex Picture Fuzzy Uncertain Linguistic information is massive modified and generalized than the Complex intuitionistic Fuzzy Uncertain Linguistic, Complex Fuzzy Uncertain Linguistic, Fuzzy Uncertain Linguistic and Uncertain Linguistic information. Keeping the supremacy and dominancy of the Complex Picture Fuzzy Uncertain Linguistic information, we investigated the Complex Picture Fuzzy Uncertain Linguistic Arithmetic Heronian Mean, Complex Picture Fuzzy Uncertain Linguistic Weighted Arithmetic Heronian Mean, Complex Picture Fuzzy Uncertain Linguistic Geometric Heronian Mean and Complex Picture Fuzzy Uncertain Linguistic Weighted Geometric Heronian Mean operators. The property of idempotency, boundedness, monotonicity, and various well-known results with certain specific cases of the invented work are also deliberated. Furthermore, in the availability of the above-proposed analysis, we constructed a multi-attribute decision-making technique by considering the diagnosed operators for complex picture fuzzy uncertain linguistic information to enhance the worth and rationality of the invented theory. Finally, we illustrated the merits and restrictions of the novel operators by comparing them with certain prevailing operators based on fuzzy generalization. Finally, in the presence of evaluated examples, we compared the pioneered operators with various existing operators to enhance the feasibility and worth of the invented operators.

Boundedness of Nonregular Pseudo-differential Operators and Their Adjoints on Variable Exponent Besov-Morrey Spaces

This paper deal with the boundedness property of non regular pseudo-differential operators a(x, D) and their adjoints a(x, D)* on variable exponent BM spaces. For this purpose, given such an operator, we use the technique of decomposition of its symbol into elementary symbols already used in other spaces.

Some Notes on Endpoint Estimates for Pseudo-differential Operators

We study the pseudo-differential operator $$\begin{aligned} T_a f\left( x\right) =\int _{\mathbb {R}^n}e^{ix\cdot \xi }a\left( x,\xi \right) \widehat{f}\left( \xi \right) \,\text {d}\xi , \end{aligned}$$where the symbol a is in the Hörmander class \(S^{m}_{\rho ,1}\) or more generally in the rough Hörmander class \(L^{\infty }S^{m}_{\rho }\) with \(m\in \mathbb {R}\) and \(\rho \in [0,1]\). It is known that \(T_a\) is bounded on \(L^1(\mathbb {R}^n)\) for \(m<n(\rho -1)\). In this paper, we mainly investigate its boundedness properties when m is equal to the critical index \(n(\rho -1)\). For any \(0\le \rho \le 1\), we construct a symbol \(a\in S^{n(\rho -1)}_{\rho ,1}\), such that \(T_a\) is unbounded on \(L^1\), and furthermore, it is not of weak type (1, 1) if \(\rho =0\). On the other hand, we prove that \(T_a\) is bounded from \(H^1\) to \(L^1\) if \(0\le \rho <1\) and construct a symbol \(a\in S^0_{1,1}\), such that \(T_a\) is unbounded from \(H^1\) to \(L^1\). Finally, as a complement, for any \(1<p<\infty \), we give an example \(a\in S^{-1/p}_{0,1}\), such that \(T_a\) is unbounded on \(L^p(\mathbb {R})\).

On The Boundedness Properties of the Generalized Fractional Integrals on the Generalized Weighted Morrey Spaces

Morrey Spaces were first introduced by C.B. Morrey in 1938. Morrey space can be considered as a generalization of the Lebesgue spaces. Morrey spaces were then generalized become the generalized Morrey spaces, the weighted Morrey spaces, and the generalized weighted Morrey spaces. One of the studies on Morrey spaces is the boundedness of certain operators on the spaces. One of the operators is the fractional integral. The boundedness of fractional integrals on the classical Morrey spaces, the weighted Morrey spaces, the generalized Morrey spaces, and the generalized weighted Morrey spaces had been known. One of the extensions of fractional integrals is generalized fractional integral. The operator was bounded on the generalized Morrey spaces. The purpose of this study is to investigate the boundedness of generalized fractional integrals on the generalized weighted Morrey spaces. The weight used is Muckenhoupt class. The results obtained show that the generalized fractional integral is bounded from generalized weighted Morrey space to another generalized weighted Morrey space under some assumptions. The main result obtained then implies the boundedness of the generalized fractional maximal operator on generalized weighted Morrey spaces under the same assumptions.

Intelligent Control of Seizure-Like Activity in a Memristive Neuromorphic Circuit Based on the Hodgkin–Huxley Model

Memristive neuromorphic systems represent one of the most promising technologies to overcome the current challenges faced by conventional computer systems. They have recently been proposed for a wide variety of applications, such as nonvolatile computer memory, neuroprosthetics, and brain–machine interfaces. However, due to their intrinsically nonlinear characteristics, they present a very complex dynamic behavior, including self-sustained oscillations, seizure-like events, and chaos, which may compromise their use in closed-loop systems. In this work, a novel intelligent controller is proposed to suppress seizure-like events in a memristive circuit based on the Hodgkin–Huxley equations. For this purpose, an adaptive neural network is adopted within a Lyapunov-based nonlinear control scheme to attenuate bursting dynamics in the circuit, while compensating for modeling uncertainties and external disturbances. The boundedness and convergence properties of the proposed control scheme are rigorously proved by means of a Lyapunov-like stability analysis. The obtained results confirm the effectiveness of the proposed intelligent controller, presenting a much improved performance when compared with a conventional nonlinear control scheme.