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Matrix Riccati Equations in Optimal Control
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作者 Malick Ndiaye 《Applied Mathematics》 2024年第3期199-213,共15页
In this paper, the matrix Riccati equation is considered. There is no general way for solving the matrix Riccati equation despite the many fields to which it applies. While scalar Riccati equation has been studied tho... In this paper, the matrix Riccati equation is considered. There is no general way for solving the matrix Riccati equation despite the many fields to which it applies. While scalar Riccati equation has been studied thoroughly, matrix Riccati equation of which scalar Riccati equations is a particular case, is much less investigated. This article proposes a change of variable that allows to find explicit solution of the Matrix Riccati equation. We then apply this solution to Optimal Control. 展开更多
关键词 Optimal Control matrix Riccati equation Change of Variable
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Numerical Methods for a Class of Quadratic Matrix Equations
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作者 GUAN Jinrui WANG Zhixin SHAO Rongxia 《应用数学》 北大核心 2024年第4期962-970,共9页
Quadratic matrix equations arise in many elds of scienti c computing and engineering applications.In this paper,we consider a class of quadratic matrix equations.Under a certain condition,we rst prove the existence of... Quadratic matrix equations arise in many elds of scienti c computing and engineering applications.In this paper,we consider a class of quadratic matrix equations.Under a certain condition,we rst prove the existence of minimal nonnegative solution for this quadratic matrix equation,and then propose some numerical methods for solving it.Convergence analysis and numerical examples are given to verify the theories and the numerical methods of this paper. 展开更多
关键词 Quadratic matrix equation M-matrix Minimal nonnegative solution Newton method Bernoulli method
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Solution of the Matrix Second Semi-Tensor Product Equation A ∘ l X ∘ l B=C
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作者 Hao Zhang 《Journal of Applied Mathematics and Physics》 2024年第10期3261-3280,共20页
In this paper, the solution of the matrix second semi-tensor product equation A∘lX∘lB=Cis studied. Firstly, the solvability of the matrix-vector second semi-tensor product equation is investigated. At the same time,... In this paper, the solution of the matrix second semi-tensor product equation A∘lX∘lB=Cis studied. Firstly, the solvability of the matrix-vector second semi-tensor product equation is investigated. At the same time, the compatibility conditions, the sufficient and necessary conditions and the specific solution methods for the matrix solution are given. Secondly, we further consider the solvability of the second semi-tensor product equation of the matrix. For each part, several examples are given to illustrate the validity of the results. 展开更多
关键词 matrix equation The Second Semi-Tensor Product Compatibility Condition Sufficient and Necessary Conditions VECTORIZATION
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An Extended Numerical Method by Stancu Polynomials for Solution of Integro-Differential Equations Arising in Oscillating Magnetic Fields
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作者 Neşe İşler Acar 《Advances in Pure Mathematics》 2024年第10期785-796,共12页
In this study, the Bernstein collocation method has been expanded to Stancu collocation method for numerical solution of the charged particle motion for certain configurations of oscillating magnetic fields modelled b... In this study, the Bernstein collocation method has been expanded to Stancu collocation method for numerical solution of the charged particle motion for certain configurations of oscillating magnetic fields modelled by a class of linear integro-differential equations. As the method has been improved, the Stancu polynomials that are generalization of the Bernstein polynomials have been used. The method has been tested on a physical problem how the method can be applied. Moreover, numerical results of the method have been compared with the numerical results of the other methods to indicate the efficiency of the method. 展开更多
关键词 Stancu Polynomials Collocation Method Integro-Differential equations Linear equation Systems matrix equations
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Matrix integrable fifth-order mKdV equations and their soliton solutions 被引量:2
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作者 马文秀 《Chinese Physics B》 SCIE EI CAS CSCD 2023年第2期47-52,共6页
We consider matrix integrable fifth-order mKdV equations via a kind of group reductions of the Ablowitz–Kaup–Newell–Segur matrix spectral problems. Based on properties of eigenvalue and adjoint eigenvalue problems,... We consider matrix integrable fifth-order mKdV equations via a kind of group reductions of the Ablowitz–Kaup–Newell–Segur matrix spectral problems. Based on properties of eigenvalue and adjoint eigenvalue problems, we solve the corresponding Riemann–Hilbert problems, where eigenvalues could equal adjoint eigenvalues, and construct their soliton solutions, when there are zero reflection coefficients. Illustrative examples of scalar and two-component integrable fifthorder mKdV equations are given. 展开更多
关键词 matrix integrable equation Riemann–Hilbert problem SOLITON
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Hermite Positive Definite Solution of the Quaternion Matrix Equation Xm + B*XB = C
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作者 Yiwen Yao Guangmei Liu +1 位作者 Yanting Zhang Jingpin Huang 《Journal of Applied Mathematics and Physics》 2023年第11期3760-3772,共13页
This paper discusses the necessary and sufficient conditions for the existence of Hermite positive definite solutions of the quaternion matrix equation X<sup>m</sup>+ B*XB = C (m > 0) and its iterative ... This paper discusses the necessary and sufficient conditions for the existence of Hermite positive definite solutions of the quaternion matrix equation X<sup>m</sup>+ B*XB = C (m > 0) and its iterative solution method. According to the characteristics of the coefficient matrix, a corresponding algebraic equation system is ingeniously constructed, and by discussing the equation system’s solvability, the matrix equation’s existence interval is obtained. Based on the characteristics of the coefficient matrix, some necessary and sufficient conditions for the existence of Hermitian positive definite solutions of the matrix equation are derived. Then, the upper and lower bounds of the positive actual solutions are estimated by using matrix inequalities. Four iteration formats are constructed according to the given conditions and existence intervals, and their convergence is proven. The selection method for the initial matrix is also provided. Finally, using the complexification operator of quaternion matrices, an equivalent iteration on the complex field is established to solve the equation in the Matlab environment. Two numerical examples are used to test the effectiveness and feasibility of the given method. . 展开更多
关键词 QUATERNION matrix equation Hermite Positive Definite Solution matrix Inequality ITERATIVE CONVERGENCE
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The Least Squares {P,Q,k+1}-Reflexive Solution to a Matrix Equation
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作者 DONG Chang-zhou LI Hao-xue 《Chinese Quarterly Journal of Mathematics》 2023年第2期210-220,共11页
Let P∈C^(m×m)and Q∈C^(n×n)be Hermitian and{k+1}-potent matrices,i.e.,P k+1=P=P∗,Qk+1=Q=Q∗,where(·)∗stands for the conjugate transpose of a matrix.A matrix X∈C m×n is called{P,Q,k+1}-reflexive(an... Let P∈C^(m×m)and Q∈C^(n×n)be Hermitian and{k+1}-potent matrices,i.e.,P k+1=P=P∗,Qk+1=Q=Q∗,where(·)∗stands for the conjugate transpose of a matrix.A matrix X∈C m×n is called{P,Q,k+1}-reflexive(anti-reflexive)if P XQ=X(P XQ=−X).In this paper,the least squares solution of the matrix equation AXB=C subject to{P,Q,k+1}-reflexive and anti-reflexive constraints are studied by converting into two simpler cases:k=1 and k=2. 展开更多
关键词 matrix equations Potent matrix {P Q k+1}-reflexive(anti-reflexive) Canonical correlation decomposition Least squares solution
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SYMMETRIC POSITIVE DEFINITE SOLUTIONS OF MATRIX EQUATIONS (AX,XB)=(C,D) AND AXB=C 被引量:1
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作者 戴华 《Transactions of Nanjing University of Aeronautics and Astronautics》 EI 1996年第2期56+52-55,共5页
The symmetric positive definite solutions of matrix equations (AX,XB)=(C,D) and AXB=C are considered in this paper. Necessary and sufficient conditions for the matrix equations to have symmetric positive de... The symmetric positive definite solutions of matrix equations (AX,XB)=(C,D) and AXB=C are considered in this paper. Necessary and sufficient conditions for the matrix equations to have symmetric positive definite solutions are derived using the singular value and the generalized singular value decompositions. The expressions for the general symmetric positive definite solutions are given when certain conditions hold. 展开更多
关键词 numerical algebra matrix equation symmetric positive definite solution
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Generally Unitary Solution to a System of Matrix Equations over the Quaternion Field
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作者 周立泰 汪远征 《Chinese Quarterly Journal of Mathematics》 CSCD 2001年第2期91-93,共3页
Generally unitary solution to the system of martix equations over the quaternion field [X mA ns =B ns ,X nn C nt =D nt ] is considered. A necessary and sufficient condition for the existence o... Generally unitary solution to the system of martix equations over the quaternion field [X mA ns =B ns ,X nn C nt =D nt ] is considered. A necessary and sufficient condition for the existence of and the expression for the generally unitary solution of the system are derived. 展开更多
关键词 generally unitary matrix system of matrix equations the quaternion field
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Minor Self-conjugate and Skewpositive Semidefinite Solutions to a System of Matrix Equations over Skew Fields
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作者 姜学波 《Chinese Quarterly Journal of Mathematics》 CSCD 2001年第2期86-90,共5页
Minor self conjugate (msc) and skewpositive semidefinite (ssd) solutions to the system of matrix equations over skew fields [A mn X nn =A mn ,B sn X nn =O sn ] are considered. Necessary and su... Minor self conjugate (msc) and skewpositive semidefinite (ssd) solutions to the system of matrix equations over skew fields [A mn X nn =A mn ,B sn X nn =O sn ] are considered. Necessary and sufficient conditions for the existence of and the expressions for the msc solutions and the ssd solutions are obtained for the system. 展开更多
关键词 minor self conjugate matrix skewpositive semidefinite matrix system of matrix equations skew field the real quatrnion field
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THE GENERALIZED REFLEXIVE SOLUTION FOR A CLASS OF MATRIX EQUATIONS (AX-B,XC=D) 被引量:7
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作者 李范良 胡锡炎 张磊 《Acta Mathematica Scientia》 SCIE CSCD 2008年第1期185-193,共9页
In this article, the generalized reflexive solution of matrix equations (AX = B, XC = D) is considered. With special properties of generalized reflexive matrices, the necessary and sufficient conditions for the solv... In this article, the generalized reflexive solution of matrix equations (AX = B, XC = D) is considered. With special properties of generalized reflexive matrices, the necessary and sufficient conditions for the solvability and the general expression of the solution are obtained. Moreover, the related optimal approximation problem to a given matrix over the solution set is solved. 展开更多
关键词 matrix equations generalized reflexive matrix optimal approximation
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SOME FURTHER NOTES ON THE MATRIX EQUATIONS A^TXB+B^TX^TA=C AND A^TXB+B^TXA=C 被引量:2
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作者 G.SOARES 《Acta Mathematica Scientia》 SCIE CSCD 2015年第1期275-280,共6页
Dehghan and Hajarian, [4], investigated the matrix equations A^TXB+B^TX^TA = C and A^TXB + B^TXA = C providing inequalities for the determinant of the solutions of these equations. In the same paper, the authors pre... Dehghan and Hajarian, [4], investigated the matrix equations A^TXB+B^TX^TA = C and A^TXB + B^TXA = C providing inequalities for the determinant of the solutions of these equations. In the same paper, the authors presented a lower bound for the product of the eigenvalues of the solutions to these matrix equations. Inspired by their work, we give some generalizations of Dehghan and Hajarian results. Using the theory of the numerical ranges, we present an inequality involving the trace of C when A, B, X are normal matrices satisfying A^T B = BA^T. 展开更多
关键词 matrix equation EIGENVALUE trace permutation matrix
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Reflexive solution to a system of matrix equations 被引量:2
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作者 常海霞 王卿文 《Journal of Shanghai University(English Edition)》 CAS 2007年第4期355-358,共4页
We derive necessary and sufficient conditions for the existence and an expression of the (anti)reflexive solution with respect to the nontrivial generalized reflection matrix P to the system of complex matrix equati... We derive necessary and sufficient conditions for the existence and an expression of the (anti)reflexive solution with respect to the nontrivial generalized reflection matrix P to the system of complex matrix equations AX = B and XC = D. The explicit solutions of the approximation problem min x∈Ф ||X - E||F was given, where E is a given complex matrix and Ф is the set of all reflexive (or antireflexive) solutions of the system mentioned above, and ||·|| is the Frobenius norm. Furthermore, it was pointed that some results in a recent paper are special cases of this paper. 展开更多
关键词 system of matrix equations Moore-Penrose inverse reflexive matrix antireflexive matrix Frobenius norm
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NEW OSCILLATION CRITERIA FOR SECOND-ORDER NONLINEAR MATRIX DIFFERENTIAL EQUATIONS 被引量:2
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作者 Xu Yancong Meng Fanwei 《Applied Mathematics(A Journal of Chinese Universities)》 SCIE CSCD 2006年第3期313-319,共7页
Some new oscillation criteria are established for the second-order matrix differential system (r(t)Z'(t))' + p(t)Z'(t) + Q(t)F(Z'(t))G(Z(t)) = 0, t ≥ to 〉 0, are different from most known ... Some new oscillation criteria are established for the second-order matrix differential system (r(t)Z'(t))' + p(t)Z'(t) + Q(t)F(Z'(t))G(Z(t)) = 0, t ≥ to 〉 0, are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of [to,∞), rather than on the whole half-line. The results weaken the condition of Q(t) and generalize some well-known results of Wong (1999) to nonlinear matrix differential equation. 展开更多
关键词 matrix differential equation OSCILLATION Riccati technique.
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HERMITE MATRIX POLYNOMIALS AND SECOND ORDER MATRIX DIFFERENTIAL EQUATIONS 被引量:6
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作者 L.Jódar R.Company 《Analysis in Theory and Applications》 1996年第2期20-30,共11页
In this paper we introduce the class of Hermite's matrix polynomials which appear as finite series solutions of second order matrix differential equations Y'-xAY'+BY=0.An explicit expression for the Hermit... In this paper we introduce the class of Hermite's matrix polynomials which appear as finite series solutions of second order matrix differential equations Y'-xAY'+BY=0.An explicit expression for the Hermite matrix polynomials,the orthogonality property and a Rodrigues' formula are given. 展开更多
关键词 exp HERMITE matrix POLYNOMIALS AND SECOND ORDER matrix DIFFERENTIAL equationS
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The General and Centro(Kewsy) Symmetric Solutions to a System of Matrix Equations over an Arbitrary Skew Field 被引量:3
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作者 QIN Jian-guo SONG Guang-ai 《Chinese Quarterly Journal of Mathematics》 CSCD 北大核心 2006年第1期66-70,共5页
Necessary and sufficient conditions are given for the existence of the general solution, the centrosymmetric solution, and the centroskewsymmetric solution to a system of linear matrix equations over an arbitrary skew... Necessary and sufficient conditions are given for the existence of the general solution, the centrosymmetric solution, and the centroskewsymmetric solution to a system of linear matrix equations over an arbitrary skew field. The representations of such the solutions of the system are also derived. 展开更多
关键词 system of matrix equations inner inverse of a matrix reflexive inverse of a matrix centro (skew) symmetric matrix skew field
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Least square method based on Haar wavelet to solve multi-dimensional stochastic Ito-Volterra integral equations
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作者 JIANG Guo KE Ting DENG Meng-ting 《Applied Mathematics(A Journal of Chinese Universities)》 SCIE CSCD 2023年第4期591-603,共13页
This paper proposes a method combining blue the Haar wavelet and the least square to solve the multi-dimensional stochastic Ito-Volterra integral equation.This approach is to transform stochastic integral equations in... This paper proposes a method combining blue the Haar wavelet and the least square to solve the multi-dimensional stochastic Ito-Volterra integral equation.This approach is to transform stochastic integral equations into a system of algebraic equations.Meanwhile,the error analysis is proven.Finally,the effectiveness of the approach is verified by two numerical examples. 展开更多
关键词 least squares method Haar wavelet Ito-Volterra integral equations integration operational matrix.
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Solutions to the generalized Sylvester matrixequations by a singular value decomposition 被引量:1
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作者 Bin ZHOU Guangren DUAN 《控制理论与应用(英文版)》 EI 2007年第4期397-403,共7页
In this paper, solutions to the generalized Sylvester matrix equations AX -XF = BY and MXN -X = TY with A, M ∈ R^n×n, B, T ∈ Rn×r, F, N ∈ R^p×p and the matrices N, F being in companion form, are est... In this paper, solutions to the generalized Sylvester matrix equations AX -XF = BY and MXN -X = TY with A, M ∈ R^n×n, B, T ∈ Rn×r, F, N ∈ R^p×p and the matrices N, F being in companion form, are established by a singular value decomposition of a matrix with dimensions n × (n + pr). The algorithm proposed in this paper for the euqation AX - XF = BY does not require the controllability of matrix pair (A, B) and the restriction that A, F do not have common eigenvalues. Since singular value decomposition is adopted, the algorithm is numerically stable and may provide great convenience to the computation of the solution to these equations, and can perform important functions in many design problems in control systems theory. 展开更多
关键词 Generalize Sylvester matrix equations General solutions Companion matrix Singular value decomposition
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The Metapositive Definite Self-Conjugate Solution of the Matrix Equation AXB=C over a Skew Field 被引量:3
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作者 王卿文 《Chinese Quarterly Journal of Mathematics》 CSCD 1995年第3期42-51,共10页
Let F be the strong p-division ring [4]. This paper is sequel to [1]. Metapositive definite self-conjugate matrix over F is defined and the necessary and sufficient conditions for determining whether a partitioned mat... Let F be the strong p-division ring [4]. This paper is sequel to [1]. Metapositive definite self-conjugate matrix over F is defined and the necessary and sufficient conditions for determining whether a partitioned matrix over F is metapositive definite self-conjugate are given.Moreover,a decomposition of pairwise matrices over F with the same numbers of columns is also presented. Whence some necessary and sufficient conditions for the existence of and the explicit expression for the metapositive definite self-conjugate solution of the matrix equation AXB=C over F are derived. 展开更多
关键词 the strong p-division ring matrix equation metapositive definite self-coniugate matrix
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Extremal ranks of the solution to a system of real quaternion matrix equations 被引量:1
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作者 俞绍文 王卿文 《Journal of Shanghai University(English Edition)》 CAS 2007年第3期229-232,共4页
In this paper, the maximal and minimal ranks of the solution to a system of matrix equations over H, the real quaternion algebra, were derived. A previous known result could be regarded as a special case of the new re... In this paper, the maximal and minimal ranks of the solution to a system of matrix equations over H, the real quaternion algebra, were derived. A previous known result could be regarded as a special case of the new result. 展开更多
关键词 system of matrix equations SOLUTION minimal rank maximal rank generalized inverse
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