Definite Solution Research Articles

ON INEXACT NEWTON METHODS FOR SOLVING TWO NONLINEAR MATRIX EQUATIONS

In this paper we consider inexact Newton methods for finding the largest positive definite solutions of two nonlinear matrix equations X+ A∗X−1A = Q and X − A∗X−1A = Q, respectively. Using Newton’s method for considered equations requires solving a Stein’s equation at each iteration. For solving the Stein’s equation, we use Smith type iterations instead of exact methods. Nonlocal convergence of the process is shown. Numerical experiments are included to illustrate the theory.

AI-Based Controls for Thermal Comfort in Adaptable Buildings: A Review

Due to global weather changes and pandemics, people are more likely to spend most of their time in indoor environments. In this regard, indoor environment quality is a very important aspect of occupant well-being, which is often ignored in modern building designs. Based on our research, thermal comfort is one of the essential items in building environments that can improve the mental stability and productivity of the occupants if the building’s indoor environment is created in a way that meets the occupants’ comfort requirements. Buildings nowadays operate on adaptive or stationary models to attain thermal comfort, which is based on Fanger’s model of the Predicted Mean Vote (PMV). Based on the literature review, limited work has been carried out to enhance the quality of the inside environment, and most research work has been devoted to building energy management. Moreover, there have been no definite solutions so far that have the capability to detect the thermal comfort requirements of multiple occupants in real time. Modern buildings tend to operate on predefined set point parameters to control the indoor environment based on the measured room temperature, which can be different from the thermal comfort requirements of the occupants. This paper discusses the limitations and assumptions that are associated with the existing thermal comfort solutions and emphasises the importance of having a real-time solution to address the thermal requirements of occupants.

Data-driven adaptive optimal control for discrete-time linear time-invariant systems

In this paper, the problem of data-driven quadratic optimal control is investigated for discrete-time linear systems without the exact knowledge of system dynamics. A value iteration-based algorithm is proposed to obtain the optimal feedback gain. The main idea of the proposed approach is to obtain two sequences for the approximation of the unique positive definite solution of the corresponding algebraic Riccati matrix equation and the optimal feedback gain by using the collected data of the system states and control inputs. The proposed algorithm could be activated by an arbitrary bounded control input and a simple initial value of the Lyapunov matrix. An important feature of this algorithm is that an original stabilising feedback gain is not required. Finally, the effectiveness of the proposed approach is verified by two examples.

An accelerated zeroing neural network for solving continuous coupled Lyapunov matrix equations

AbstractIn this paper, an improved zeroing neural network (ZNN) model is proposed to obtain the positive definite solutions of the continuous coupled Lyapunov matrix equations (CLMEs) associated with continuous‐time Markovian jump (CMJ) systems. To achieve this, a general ZNN model is established by constructing a matrix‐valued error function. Then, to accelerate the convergence rate of the proposed ZNN model, the latest estimation is introduced to obtain an improved ZNN model. Some convergence conditions have been derived for the presented improved ZNN model through Lyapunov theory. Comparisons among the improved ZNN model and the existing results are conducted to illustrate the advantages of the proposed improved ZNN model in numerical examples.

Data‐driven adaptive optimal control for discrete‐time periodic systems

AbstractIn this paper, a problem of data‐driven optimal control is studied for discrete‐time periodic systems with unknown system matrices and input matrices. For this problem, a value iteration‐based adaptive dynamic programming algorithm is proposed to obtain the suboptimal controller. The core of the algorithm proposed in this paper is to obtain an approximation of the unique positive definite solution of the algebraic Riccati equation and the optimal feedback gain matrix by using the collected real‐time data of the system states and control inputs. Without an initial stabilizing feedback gain, the proposed algorithm could be activated by an arbitrary bounded control input. Finally, the effectiveness of the proposed approach is demonstrated by two examples.

Bank Interest in the Contemporary Era: Problem of Ad'afan Muda'afah Interpretation in Determining Law of Usury

Usury in Islam is a prohibited system in the Qur'an and hadith. The system is not recognized in the modern banking system, leading to different opinions among Muslims in determining the legality of bank interest. Therefore, this research aimed to explore the contemporary law of bank interest, focusing on the different opinions among the mufasir on the definition of Ad'afan Muda'afah as a basis for determining the legality of usury. A normative method was adopted, and a literature review of several scholars was conducted on usury law and the interpretation of Ad'afan Muda'afah. The data was analyzed considering the modern banking system within the framework of Islamic law, particularly Maqashid Shari'ah. The results showed that the difference in the illat used for bank interest law varied in legal conclusions. Despite efforts to forbid bank interest, Muslims did not attain a definite solution to avoid this issue. Consequently, bank interest was temporarily sanctioned under certain conditions such as not being compounded, having low interest, and not comprising any form of persecution. To address this problem, Muslims should support the existence of Islamic bank producing muamalah products according to Islamic Sharia.

String Viscous Fluid Cosmological Model in the Framework of Sen-Dunn Theory of Gravitation

Objective: To present a new solution to the field equations for higher-dimensional Bianchi type-III string viscous fluid cosmological model in the context of the Sen-Dunn theory of gravity. Methods: To obtain definite solution of the field equations, we consider a power law relation between the scale factor and scalar field, and we take into account two distinct scale factors, and , in which , are positive constants. This produces a time-dependent deceleration parameter. We analyse the model with constant bulk viscosity constant to explain the current accelerated expansion of the universe. Findings: Spatial volume of the model increases with cosmic time, which shows that universe is spatially expanding. Also, the model possesses a point-type singularity. It is noted that these models depict the universe's accelerated expansion. Novelty: We obtained new solution to the field equations for higher dimensional Bianchi type-III generated by means of a cloud of strings with bulk viscous fluid in Sen-Dunn theory by using quadratic form. Keywords: Bianchi type-III, Cosmic string, Bulk viscosity, Sen-Dunn Theory

Supplier Evaluation in Supply Chain Environment Based on Radial Basis Function Neural Network

The comprehensive evaluation and selection of suppliers under the environment of supply chain management has become a key factor affecting the success of supply chain. How to select suppliers and the strategic partnership between suppliers under the environment of supply chain management has become an important challenge. To solve this problem, this paper takes the supplier evaluation and selection of Guangzhou Automobile Toyota Company as the research object, constructs the index system of supplier comprehensive evaluation and selection, uses the RBF neural network algorithm to establish the supplier evaluation and selection model, and makes an experimental study. The results show that radial basis function neural network is a local approximation network, which has a unique and definite solution to the problem, and there is no local minimum problem in BP network. It is a method that enables enterprises and suppliers to have a clear understanding and seek further promotion together. The research provides theoretical data support for enterprise managers to make decisions.

Research on precise and standardized numerical simulation strategy for vehicle aerodynamics

PurposeThe purpose of this study is to propose a precise and standardized strategy for numerically simulating vehicle aerodynamics.Design/methodology/approachError sources in computational fluid dynamics were analyzed. Additionally, controllable experiential and discretization errors, which significantly influence the calculated results, are expounded upon. Considering the airflow mechanism around a vehicle, the computational efficiency and accuracy of each solution strategy were compared and analyzed through numerous computational cases. Finally, the most suitable numerical strategy, including the turbulence model, simplified vehicle model, calculation domain, boundary conditions, grids and discretization scheme, was identified. Two simplified vehicle models were introduced, and relevant wind tunnel tests were performed to validate the selected strategy.FindingsErrors in vehicle computational aerodynamics mainly stem from the unreasonable simplification of the vehicle model, calculation domain, definite solution conditions, grid strategy and discretization schemes. Using the proposed standardized numerical strategy, the simulated steady and transient aerodynamic characteristics agreed well with the experimental results.Originality/valueBuilding upon the modified Low-Reynolds Number k-e model and Scale Adaptive Simulation model, to the best of the authors’ knowledge, a precise and standardized numerical simulation strategy for vehicle aerodynamics is proposed for the first time, which can be integrated into vehicle research and design.

A unified study of the redundant control input problem with different conditions in the stabilizable case

This paper investigates the redundant control input problem in the sense of linear quadratic regulator optimal control. Increasing redundant control inputs leads to a decrease in the quadratic performance index. However, in some cases, this also increases the norm of the feedback matrix, resulting in actuator saturation. To solve the problem, a comparative analysis of the latest research results on the redundant control input problem under different conditions is first presented. Secondly, new tighter upper bounds of the positive (semi-)definite solution of the continuous algebraic Riccati equation (CARE) corresponding to the redundant control input problem are studied uniformly in the stabilizable case. Thirdly, based on them, a class of sufficient conditions to guarantee that the norm of the feedback matrix decreases is presented. Finally, numerical experiments demonstrate the effectiveness and superiority of our conclusions. Meanwhile, a simulation experiment demonstrates that systems with redundant control inputs can achieve good control performance.

High-performance LQR control for continuous time systems with redundant control inputs

This paper investigates the linear quadratic regulator (LQR) problem for continuous-time linear systems with redundant control inputs. The LQR control problem has been widely studied, but due to technical difficulties, there is little advance in the study of adding input redundancies. To address this problem, a convergent matrix series, the limit of a monotonically decreasing sequence, is first presented as an upper bound of the symmetric positive definite solution of a class of continuous algebraic Riccati equations (CAREs). Compared with the existing studies on this topic, the obtained bounds are more precise. Moreover, by utilising these upper bounds in the LQR problem of adding input redundancies, a class of sufficient conditions is presented to decrease the quadratic performance index. Finally, the corresponding numerical examples are given to illustrate the effectiveness of our results and compare them with the existing results. Meanwhile, some simulation experiments show that systems with redundant control inputs have good control performance.

Modified Riccati iterative algorithms for stochastic coupled algebraic Riccati equations of linear stochastic Markovian jump systems

AbstractThis paper investigates the stochastic coupled algebraic Riccati equations (SCAREs) associated with linear stochastic Markovian jump systems. First, two existing algorithms utilized to solve the SCAREs are reviewed and discussed. Second, based on the analysis of these two algorithms, a modified Riccati iterative algorithm is proposed with a tuning parameter added. It is verified that the proposed algorithm with zero initial values converges to the unique positive definite solution of the SCAREs with fast speed. Furthermore, two numerical examples are given to verify the efficiency and superiority of the proposed algorithm.

Suzuki-type fixed point theorems in relational metric spaces with applications

In this paper, we establish a relation-theoretic version of the results presented by Kim et al. (Journal of Nonlinear and Convex Analysis, 16 (9) (2015), 1779-1786). To showcase the versatility of our results, we furnish some illustrative examples. Furthermore, we exhibit an application of our results to establish sufficient conditions for the existence of a positive definite common solution to a pair of nonlinear matrix equations.

Pricing geometric average Asian options in the mixed sub-fractional Brownian motion environment with Vasicek interest rate model

<p>Considering the characteristics of long-range correlations in financial markets, the issue of valuing geometric average Asian options is examined, assuming that the variations of the underlying asset follow the mixed sub-fractional Brownian motion, and the dynamics of short-term interest rate satisfies the mixed sub-fractional Vasicek model. Based on the principle of no arbitrage, the definite solution of PDE of a zero-coupon bond for geometric average Asian options under the circumstance of the mixed sub-fractional is given by the delta hedging technique. The derivation of the explicit pricing formula for geometric average Asian options with fixed strike price is achieved through the utilization of multiple variable substitutions. Furthermore, we perform numerical calculations to analyze the performance of the model.</p>

Solving nonlinear matrix and Riesz-Caputo fractional differential equations via fixed point theory in partial metric spaces

A modified implicit relation and ?-implicit contractive condition are introduced in the setting of relational partial metric spaces and some related fixed point results are derived. Two suitable examples are provided. As an application, sufficient conditions are derived for the existence of a unique positive definite solution of the non-linear matrix equation X = B + Pk i=1A* i T(X)Ai. An example is given, using matrices that are randomly generated, as well as convergence and error analysis and average CPU time analysis. Solving fractional differential equations of Riesz-Caputo type with anti-periodic boundary conditions is also discussed, followed by two illustrations.

Linear solution of the passive localization problem with combined single-station direction finding and two-station time difference information

Based on the main station’s direction measurement value and the time difference measurement value of the two stations, two definite solution equations can be listed in the polar coordinate system by directly utilizing the path difference equation and the cosine theorem, and the analytical solution can be obtained by solving the two definite solution equations jointly. Further comparison of the ranging errors shows that the ranging accuracy of the two-station directional and time-difference positioning system is only slightly worse than that of the three-station time-difference positioning system. The new results provide a better basis for the engineering design of two-station hybrid localization.

On positive definite solutions of the matrix equation $ X-\sum_{i = 1}^{m}A_{i}^{\ast}X^{-p_{i}}A_{i} = Q $

<p>In this paper, we used the outstanding properties of the Thompson metric to conclusively demonstrate the existence of a unique positive definite solution for the nonlinear matrix equation $ X-\sum_{i = 1}^{m}A_{i}^{\ast}X^{-p_{i}}A_{i} = Q $ without any additional assumptions. Furthermore, we designed an iterative algorithm to compute this unique positive definite solution, and derive its corresponding error estimate formula. Additionally, we presented three refined existence intervals for positive definite solutions of this equation. Finally, numerical examples were employed to validate the practicability of our iterative algorithm.</p>

Solution Approach to Cutting Stock Problems Using Iterative Trim Loss Algorithm and Monte-Carlo Simulation

Cutting Stock Problems are the most studied NP-Hard combinatorial problems among optimization problems. An One-dimensional Cutting Stock Problem (1-CSP), which aims to create cutting patterns to minimize trim loss, is one of the best known optimization problems. The difficulty of the solution stages and the lack of a definite solution method that can be applied to all problems have caused these problems to attract a lot of attention by researchers. This study includes a hybrid solution algorithm combined with iterative trim loss algorithm and Monte Carlo simulations, and a comparative study of the method with the solution methods in the literature, for the solution of orders to be obtained with minimum cutting loss from the same type of stocks.

A new inversion-free iterative algorithm for the discrete algebraic Riccati equation

Abstract In this paper, by the transformation form of the discrete algebraic Riccati equation (DARE), we propose a new inverse-free iterative algorithm to obtain the positive definite solution of the DARE. Furthermore, the monotone convergence is proved and convergence rate analysis is presented for the derived algorithm. Compared with some existing algorithms, numerical examples demonstrate the feasibility and effectiveness of our algorithm.

Positive Definite Solution of a Class of Matrix Equations

Positive Definite Solution of a Class of Matrix Equations