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ESTIMATES FOR EIGENVALUES OF NEUMANN AND NAVIER PROBLEM
Deng, Yanlin,Du, Feng,Hou, Lanbao Korean Mathematical Society 2021 대한수학회보 Vol.58 No.6
In this paper, we firstly prove some general inequalities for the Neumann eigenvalues for domains contained in a Euclidean n-space ℝ<sup>n</sup>. Using the general inequalities, we can derive some new Neumann eigenvalues estimates which include an upper bound for the (k + 1)<sup>th</sup> eigenvalue and a new estimate for the gap of the consecutive eigenvalues. Moreover, we give sharp lower bound for the first eigenvalue of two kinds of eigenvalue problems of the biharmonic operator with Navier boundary condition on compact Riemannian manifolds with boundary and positive Ricci curvature.
An inequality involving the second largest and smallest eigenvalue of a distance-regular graph
Koolen, Jack H.,Park, Jongyook,Yu, Hyonju Elsevier 2011 Linear algebra and its applications Vol.434 No.12
<P><B>Abstract</B></P><P>For a distance-regular graph with second largest eigenvalue (resp., smallest eigenvalue) <SUB>θ1</SUB> (resp., <SUB>θD</SUB>) we show that (<SUB>θ1</SUB>+1)(<SUB>θD</SUB>+1)⩽-<SUB>b1</SUB> holds, where equality only holds when the diameter equals two. Using this inequality we study distance-regular graphs with fixed second largest eigenvalue.</P>
Chen, Su Huan,Song, Min,Chen, Yu Dong Techno-Press 2005 Structural Engineering and Mechanics, An Int'l Jou Vol.21 No.2
Variations in system parameters due to uncertainties may result in system performance deterioration. Uncertainties in modeling of structures are often considered to ensure that control system is robust with respect to response errors. Hence, the uncertain concept plays an important role in vibration control of the engineering structures. The paper discusses the robustness of the stability of vibration control systems with uncertain parameters. The vibration control problem of an uncertain system is approximated by a deterministic one. The uncertain parameters are described by interval variables. The uncertain state matrix is constructed directly using system physical parameters and avoided to use bounds in Euclidean norm. The feedback gain matrix is determined based on the deterministic systems, and then it is applied to the actual uncertain systems. A method to calculate the upper and lower bounds of eigenvalues of the close-loop system with uncertain parameters is presented. The lower bounds of eigenvalues can be used to estimate the robustness of the stability the controlled system with uncertain parameters. Two numerical examples are given to illustrate the applications of the present approach.
Lower Solution Bounds of the Continuous Coupled Algebraic Riccati Matrix Equation
Jianzhou Liu,Juan Zhang 제어·로봇·시스템학회 2012 International Journal of Control, Automation, and Vol.10 No.6
In this paper, applying eigenvalue sum inequality of symmetric matrix and the properties of M-matrix and its inverse matrix, we introduce new lower matrix bounds for the solution of the continuous coupled algebraic Riccati equation. Finally, we give corresponding numerical examples to demonstrate the effectiveness of the derived results.
Robust Fuzzy Controller Design for Dynamic Positioning System of Ships
Werneld Egno Ngongi,Jialu Du,Rui Wang 제어·로봇·시스템학회 2015 International Journal of Control, Automation, and Vol.13 No.5
This paper presents a robust fuzzy controller design approach for dynamic positioning (DP) system of ships using optimal H∞ control techniques. The H∞ control technique is used to exterminate the effects of environmental disturbances. Firstly, a Takagi-Sugeno (TS) fuzzy model is applied to approximate the nonlinear DP system. Next, linear matrix inequality (LMI) and general eigenvalue problem (GEVP) methods are employed to find a positive definite matrix and controller gains. The stability of the controller is proven by using Lyapunov stability theorems. A positive definite matrix is determined by solving LMI equations using robust control toolbox available in MATLAB. The obtained positive definite matrix proves that the designed fuzzy controller is stable. Finally, a uniformly ultimately bound (UUB) and control performance for the dynamic position system is guaranteed. Simulation is carried out, and results are presented to validate the effectiveness and performance of the proposed control system.
New Upper Matrix Bounds for the Solution of the Continuous Algebraic Riccati Matrix Equation
Richard Keith Davies,Peng Shi,Ron Wiltshire 대한전기학회 2008 International Journal of Control, Automation, and Vol.6 No.5
In this paper, new upper matrix bounds for the solution of the continuous algebraic Riccati equation (CARE) are derived. Following the derivation of each bound, iterative algorithms are developed for obtaining sharper solution estimates. These bounds improve the restriction of the results proposed in a previous paper, and are more general. The proposed bounds are always calculated if the stabilizing solution of the CARE exists. Finally, numerical examples are given to demonstrate the effectiveness of the present schemes.
Jianzhou Liu,Juan Zhang 제어·로봇·시스템학회 2012 International Journal of Control, Automation, and Vol.10 No.6
In this paper, if the coefficient matrices in the continuous coupled algebraic Riccati equation (CCARE) undergo perturbations, with the aid of the equivalent form for the perturbation of the CCARE and the classical eigenvalue inequalities, we observe new upper matrix bounds for the perturbation of the CCARE through solving the linear inequalities. Finally, we present corresponding numerical examples to show the effectiveness of the derived results.
The Improved Upper Solution Bounds of the Continuous Coupled Algebraic Riccati Matrix Equation
Jianzhou Liu,Juan Zhang 제어·로봇·시스템학회 2013 International Journal of Control, Automation, and Vol.11 No.4
In this paper, combining some special eigenvalue inequalities of matrix's product and sum with the equivalent form of the continuous coupled algebraic Riccati equation (CCARE), we construct linear inequalities. Then, in terms of the properties of M-matrix and its inverse matrix, through solving the derived linear inequalities, we offer new upper matrix bounds for the solution of the CCARE, which improve some of the recent results. Finally, we present a corresponding numerical example to show the effectiveness of the given results.
SPECTRAL ANALYSIS OF THE MGSS PRECONDITIONER FOR SINGULAR SADDLE POINT PROBLEMS
RAHIMIAN, MARYAM,SALKUYEH, DAVOD KHOJASTEH The Korean Society for Computational and Applied M 2020 Journal of applied mathematics & informatics Vol.38 No.1
Recently Salkuyeh and Rahimian in (Comput. Math. Appl. 74 (2017) 2940-2949) proposed a modification of the generalized shift-splitting (MGSS) method for solving singular saddle point problems. In this paper, we present the spectral analysis of the MGSS preconditioner when it is applied to precondition the singular saddle point problems with the (1, 1) block being symmetric. Some eigenvalue bounds for the spectrum of the preconditioned matrix are given. We show that all the real eigenvalues of the preconditioned matrix are in a positive interval and all nonzero eigenvalues having nonzero imaginary part are contained in an intersection of two circles.
SPECTRAL ANALYSIS OF THE MGSS PRECONDITIONER FOR SINGULAR SADDLE POINT PROBLEMS
Maryam Rahimian,Davod Khojasteh Salkuyeh 한국전산응용수학회 2020 Journal of applied mathematics & informatics Vol.38 No.1
Recently Salkuyeh and Rahimian in (Comput. Math. Appl. 74 (2017) 2940-2949) proposed a modification of the generalized shift-splitting (MGSS) method for solving singular saddle point problems. In this paper, we present the spectral analysis of the MGSS preconditioner when it is applied to precondition the singular saddle point problems with the (1,1) block being symmetric. Some eigenvalue bounds for the spectrum of the preconditioned matrix are given. We show that all the real eigenvalues of the preconditioned matrix are in a positive interval and all nonzero eigenvalues having nonzero imaginary part are contained in an intersection of two circles.