Nonnegative Definite Solutions Research Articles

Unique non-negative definite solution of the time-varying algebraic Riccati equations with applications to stabilization of LTV systems

In the context of infinite-horizon optimal control problems, the algebraic Riccati equations (ARE) arise when the stability of linear time-varying (LTV) systems is investigated. Using the zeroing neural network (ZNN) approach to solve the time-varying eigendecomposition-based ARE (TVE-ARE) problem, the ZNN model (ZNNTVE-ARE) for solving the TVE-ARE problem is introduced as a result of this research. Since the eigendecomposition approach is employed, the ZNNTVE-ARE model is designed to produce only the unique nonnegative definite solution of the time-varying ARE (TV-ARE) problem. It is worth mentioning that this model follows the principles of the ZNN method, which converges exponentially with time to a theoretical time-varying solution. The ZNNTVE-ARE model can also produce the eigenvector solution of the continuous-time Lyapunov equation (CLE) since the Lyapunov equation is a particular case of ARE. Moreover, this paper introduces a hybrid ZNN model for stabilizing LTV systems in which the ZNNTVE-ARE model is employed to solve the continuous-time ARE (CARE) related to the optimal control law. Experiments show that the ZNNTVE-ARE and HFTZNN-LTVSS models are both effective, and that the HFTZNN-LTVSS model always provides slightly better asymptotic stability than the models from which it is derived.

Nonnegative definite and Re-nonnegative definite solutions to a system of matrix equations with statistical applications

Necessary and sufficient conditions are given for the existence of a nonnegative definite solution, a Re-nonnegative definite solution, a positive definite solution and a Re-positive definite solution to the system of matrix equationsAXA*=CandBXB*=D,respectively. The expressions for these special solutions are given when the consistent conditions are satisfied. Based on the new results, the characterization of the covariance matrix such that a pair of multivariate quadratic forms are distributed as independent noncentral Wishart random matrices is derived. Many results existing in the literature are extended.

Hermitian and non-negative definite reflexive and anti-reflexive solutions to AX=B

ABSTRACTIn this paper, we establish some conditions for the existence and the representations for the Hermitian reflexive and Hermitian anti-reflexive, and non-negative definite reflexive solutions to the matrix equation AX=B with respect to a generalized reflection matrix P by using the Moore–Penrose inverse. Moreover, in corresponding solution sets of the equation, the explicit expressions of the nearest matrix to a given matrix in the Frobenius norm have been provided.

A General Hermitian Nonnegative-Definite Solution to the Matrix Equation <i>AXB</i> = <i>C</i>

We derive necessary and sufficient conditions for the existence of a Hermitian nonnegative-definite solution to the matrix equation AXB = C. Moreover, we derive a representation of a general Hermitian nonnegative-definite solution. We then apply our solution to two examples, including a comparison of our solution to a proposed solution by Zhang in [1] using an example problem given from [1]. Our solution demonstrates that the proposed general solution from Zhang in [1] is incorrect. We also give a second example in which we derive the general covariance structure so that two matrix quadratic forms are independent.

Solvability of systems of linear matrix equations subject to a matrix inequality

In this paper, the solvability conditions and the explicit expressions of the Hermitian solutions to the system of matrix equationsand the Hermitian nonnegative definite solutions to the system of matrix equationsare, respectively, put forward, by making full use of the generalized inverse and the rank of matrices. As applications, some special cases of the above systems of matrix equations are considered. In addition, the maximal ranks and inertias of the Hermitian solutions are, respectively, presented.

Discrete-time mean-field Stochastic linear–quadratic optimal control problems, II: Infinite horizon case

This paper first presents results on the equivalence of several notions of L2-stability for linear mean-field stochastic difference equations with random initial value. Then, it is shown that the optimal control of a mean-field linear–quadratic optimal control with an infinite time horizon uniquely exists, and the optimal control can be expressed as a linear state feedback involving the state and its mean, via the minimal nonnegative definite solution of two coupled algebraic Riccati equations. As a byproduct, the open-loop L2-stabilizability is proved to be equivalent to the closed-loop L2-stabilizability. Moreover, the minimal nonnegative definite solution, the maximal solution, the stabilizing solution of the algebraic Riccati equations and their relations are carefully investigated. Specifically, it is shown that the maximal solution is employed to construct the optimal control and value function to another infinite time horizon mean-field linear–quadratic optimal control. In addition, the maximal solution being the stabilizing solution, is completely characterized by properties of the coefficients of the controlled system. This enriches the existing theory about stochastic algebraic Riccati equations. Finally, the notion of exact detectability is introduced with its equivalent characterization of stochastic versions of the Popov–Belevitch–Hautus criteria. It is then shown that the minimal nonnegative definite solution is the stabilizing solution if and only if the uncontrolled system is exactly detectable.

The Hermitian and nonnegative definite solutions of AX=B subject to CXC^* ≥ D

The Hermitian and nonnegative definite solutions of AX=B subject to CXC^* ≥ D

THE EXTREMAL RANKS AND INERTIAS OF THE LEAST SQUARES SOLUTIONS TO MATRIX EQUATION AX = B SUBJECT TO HERMITIAN CONSTRAINT

In this paper, the formulas for calculating the extremal ranks and inertias of the Hermitian least squares solutions to matrix equation AX = B are established. In particular, the necessary and sufficient conditions for the existences of the positive and nonnegative definite solutions to this matrix equation are given. Meanwhile, the least squares problem of the above matrix equation with Hermitian R-symmetric and R-skew symmetric constraints are also investigated.

Common nonnegative definite solutions of some classical matrix equations

A necessary and sufficient condition for the existence of a Hermitian nonnegative definite solution of system of matrix equations $$A_{1}X=C_{1},\qquad XB_{2}=C_{2}, \qquad A_{3}XA_{3}^{\ast}=C_{3},\qquad A_{4}XA_{4}^{\ast}=C_{4} $$ as well as a representation for this general nonnegative definite solution are derived. As particular cases, the corresponding results on some other systems are also derived.

A Hermitian least squares solution of the matrix equation [formula omitted] subject to inequality restrictions

This paper gives some closed-form formulas for computing the maximal and minimal ranks and inertias of P−X with respect to X, where P∈CHn is given, and X is a Hermitian least squares solution to the matrix equation AXB=C. We derive, as applications, necessary and sufficient conditions for X⩾(⩽,>,<)P in the Löwner partial ordering. In addition, we give necessary and sufficient conditions for the existence of a Hermitian positive (negative, nonpositive, nonnegative) definite least squares solution to AXB=C.

The common bisymmetric nonnegative definite solutions with extreme ranks and inertias to a pair of matrix equations

We in this paper consider the bisymmetric nonnegative definite solution with extremal ranks and inertias to a system of quaternion matrix equations AX=C, XB=D. We derive the extremal ranks and inertias of the common bisymmetric nonnegative definite solution to the system. The general expressions of the bisymmetric nonnegative definite solution with extremal ranks and inertias to the system mentioned above are also presented. In addition, we give a numerical example to illustrate the results of this paper.

On Hermitian nonnegative-definite solutions to matrix equations

In this note, for a system of q matrix equations of the form $$ A_i XA_i^* = B_i B_i^* ,i = 1,2,...,q, $$ we consider the problem of the existence of Hermitian nonnegative-definite solutions. We offer an alternative with simplification and regularity to the result on necessary and sufficient conditions for the above matrix equations with q = 2 to have a Hermitian nonnegative-definite solution obtained by Zhang [1], who proposed a revised version of Young et al. [2]. Moreover, we give a necessary condition for the general case and then put forward a conjecture, with which at least some special situations are in agreement.

Nonnegative definite solutions to matrix equations with applications to multivariate test statistics

LetX ∼N p, n (M, Σ), where the meanM is a matrix of orderp × n and the covariance matrixΣ is a nnd of orderp n. In this paper we first obtain a version of Cochran’s theorem. Basically, the theorem reduces the problem of verifying Wishartness and independence of matrix quadratic forms inX to solving matrix equations inΣ. Next, we obtain characterizations of the class of nnd solutionsΣ to those matrix equations. As an application we give a simple description of the class of nnd matrices such that the distributions of common multivariate test statistics are invariant except for a scale factor.

On rank-constrained Hermitian nonnegative-definite least squares solutions to the matrix equation AXA H =B

In the literature, rank-constrained Hermitian nonnegative-definite solutions to the matrix equation AXA H =B have been investigated, under the conditions that B is Hermitian and nonnegative-definite, and the matrix equation is consistent. In this paper, we discuss rank-constrained Hermitian nonnegative-definite least squares solutions to this matrix equation, in which the above conditions may not hold. We derive the rank range and expression of these least squares solutions. Therefore, the results obtained in this paper generalize those in the literature.

A characterization of solutions of the discrete-time algebraic Riccati equation based on quadratic difference forms

This paper is concerned with a characterization of all symmetric solutions to the discrete-time algebraic Riccati equation (DARE). Dissipation theory and quadratic difference forms from the behavioral approach play a central role in this paper. Along the line of the continuous-time results due to Trentelman and Rapisarda [H.L. Trentelman, P. Rapisarda, Pick matrix conditions for sign-definite solutions of the algebraic Riccati equation, SIAM J. Contr. Optim. 40 (3) (2001) 969–991], we show that the solvability of the DARE is equivalent to a certain dissipativity of the associated discrete-time state space system. As a main result, we characterize all unmixed solutions of the DARE using the Pick matrix obtained from the quadratic difference forms. This characterization leads to a necessary and sufficient condition for the existence of a non-negative definite solution. It should be noted that, when we study the DARE and the dissipativity of the discrete-time system, there exist two difficulties which are not seen in the continuous-time case. One is the existence of a storage function which is not a quadratic function of state. Another is the cancellation between the zero and infinite singularities of the dipolynomial spectral matrix associated with the DARE, due to the infinite generalized eigenvalues of the associated Hamiltonian pencil. One of the main contributions of this paper is to demonstrate how to resolve these difficulties.

The general common Hermitian nonnegative-definite solution to the matrix equations [formula omitted] and [formula omitted] with applications in statistics

We deduce a necessary and sufficient condition for the matrix equations AXA ∗=BB ∗ and CXC ∗=DD ∗ to have a common Hermitian nonnegative-definite solution and a representation of the general common Hermitian nonnegative-definite solution to these two equations when they have such common solutions. Thereby, we solve a statistical problem which is concerned in testing linear hypotheses about regression coefficients in the multivariate linear model. This paper is a revision of Young et al. (J. Multivariate Anal. 68 (1999) 165) whose mistake was pointed out in (Linear Algebra Appl. 321 (2000) 123).

The general common Hermitian nonnegative-definite solution to the matrix equations AXA*=BB* and CXC*=DD* with applications in statistics

We deduce a necessary and sufficient condition for the matrix equations AXA ∗ =BB ∗ and CXC ∗ =DD ∗ to have a common Hermitian nonnegative-definite solution and a representation of the general common Hermitian nonnegative-definite solution to these two equations when they have such common solutions. Thereby, we solve a statistical problem which is concerned in testing linear hypotheses about regression coefficients in the multivariate linear model. This paper is a revision of Young et al. (J. Multivariate Anal. 68 (1999) 165) whose mistake was pointed out in (Linear Algebra Appl. 321 (2000) 123).

The general common Hermitian Nonnegative-definite solution to the matrix equations AXA*=B and CXC*=D

We give a necessary and sufficient condition for the existence of at least one common Hermitian nonnegative-definite (positive-definite) solution to the pair of matrix equations AXA*=B and CXC*=D, and derive a representation of the general common Hermitian nonnegative-definite (positive-definite) solution to these two equations when they have at least such one common solution. This article can be viewed as a supplementary version of that derived by Young et al. (D.M. Young, J.W. Seaman Jr. and L.M. Meaux (1999). Independence distribution preserving covariance structures for themultivariate model. J. Multivariate Anal., 68, 165–175) since Groß (J. Groß (2000). Nonnegative-define and positive solutions to the matrix equation AXA*=B – revisited. Linear Algebra Appl., 321, 123–129) has given a counterexample to point out a mistake in their basic Theorem 1. The proposed approach is convenient to use and possesses good numerical reliability since it mainly involves only a series of singular value decompositions and the inverses of several positive definite diagonal matrices.

The Kleinman Iteration for Nonstabilizable Systems

We consider the computation of Hermitian nonnegative definite solutions of algebraic Riccati equations. These solutions are the limit, P=limi →∞ Pi, of a sequence of matrices obtained by solving a sequence of Lyapunov equations. The procedure parallels the well-known Kleinman technique but the stabilizability condition on the underlying linear time-invariant system is removed. The convergence of the constructed sequence {Pi }i≥1 is guaranteed by the minimality of Pi in the set of Hermitian nonnegative definite solutions of the Lyapunov equation in the ith iteration step.

Linear-Quadratic Optimal Control Problems for Discrete Descriptor Systems

The control in the feedback form and the optimal cost are obtained for four linear-quadratic control problems by discrete descriptor systems in a Hilbert space. Special solutions of the implicit operator Riccati equations used here act in all state space and in general case they are not non-negative definite unlike the operators in the subspace and the non-negative definite solutions, acting in all state space, as it was in other authors papers. The examples show that the considered operator Riccati equations have no non-negative definite solutions though the corresponding optimal control problems have the unique solutions.