On the discrete and continuous time infinite-dimensional algebraic Riccati equations

The standard state space solution of the finite-dimensional continuous time quadratic cost minimization problem has a straightforward extension to infinite-dimensional problems with bounded or moderately unbounded control and observation operators. However, if these operators are allowed to be sufficiently unbounded, then a strange change takes place in one of the coefficients of the algebraic Riccati equation, and the continuous time Riccati equation begins to resemble the discrete time Riccati equation. To explain why this phenomenon must occur we discuss a particular hyperbolic PDE in one space dimension with boundary control and observation (a transmission line) that can be formulated both as a discrete time system and as a continuous time system, and show that in this example the continuous time Riccati equation can be recovered from the discrete time Riccati equation. A particular feature of this example is that the Riccati operator does not map the domain of the generator into the domain of the adjoint generator, as it does in the standard case.     

Lower bounds on the solution of Lyapunov matrix and algebraic Riccati equations

In this note various lower bounds for all the eigenvalues of the solution matrixKof the Lyapunov matrix equation are established. A special case of this result is a generalization of that presented in [1]-[3], where lower bounds for the maximum and minimum eigenvalues ofKare given. Moreover, the approach used here enables one to establish various lower bounds for some of the (largest) eigenvalues of the solution matrix of the algebraic Riccati equation.     

Upper solution bounds of the continuous coupled algebraic Riccati matrix equation

In this article, by using some matrix identities, we construct the equivalent form of the continuous coupled algebraic Riccati equation (CCARE). Further, with the aid of the eigenvalue inequalities of matrix's product, by solving the linear inequalities utilising the properties of M-matrix and its inverse matrix, new upper matrix bounds for the solutions of the CCARE are established, which improve and extend some of the recent results. Finally, a corresponding numerical example is proposed to illustrate the effectiveness of the derived results.     

Monotonicity of maximal solutions of algebraic Riccati equations

A monotonicity result for the maximal solution of the equation XBB^*X − A^*X − XA − Q = 0, Q = Q^*, (A, B) stabilizable, is proved.     

Eigenvalue decay bounds for solutions of Lyapunov equations: the symmetric case

We present two new bounds for the eigenvalues of the solutions to a class of continuous- and discrete-time Lyapunov equations. These bounds hold for Lyapunov equations with symmetric coefficient matrices and right-hand side matrices of low rank. They reflect the fast decay of the nonincreasingly ordered eigenvalues of the solution matrix.     

Upper solution bounds of the continuous and discrete coupled algebraic Riccati equations

In this paper, we propose upper bounds for the sum of the maximal eigenvalues of the solutions of the continuous coupled algebraic Riccati equation (CCARE) and the discrete coupled algebraic Riccati equation (DCARE), which are then used to infer upper bounds for the maximal eigenvalues of the solutions of each Riccati equation. By utilizing the upper bounds for the maximal eigenvalues of each equation, we then derive upper matrix bounds for the solutions of the CCARE and DCARE. Following the development of each bound, an iterative algorithm is proposed which can be used to derive tighter upper matrix bounds. Finally, we give numerical examples to demonstrate the effectiveness of the proposed results, making comparisons with existing results.     

Bounds for the eigenvalues of the solution of the unified algebraic Riccati matrix equation

In recent years, several eigenvalues bounds have been investigated separately for the solutions of the continuous and the discrete Riccati and Lyapunov matrix equations. In this paper, lower bounds for the eigenvalues of the solution of the unified Riccati equation (relatively to continuous and discrete cases), are presented. In the limiting cases, the results reduce to some new bounds for both the continuous and discrete Riccati equation.     

Strong solutions of the discrete-time algebraic Riccati equation

Conditions are given under which a solution of the DARE is positive semidefinite if and only if all the eigenvalues of its associated closed-loop matrix are in the closed unit disc.     

New matrix bounds and iterative algorithms for the discrete coupled algebraic Riccati equation

The discrete coupled algebraic Riccati equation (DCARE) has wide applications in control theory and linear system. In general, for the DCARE, one discusses every term of the coupled term, respectively. In this paper, we consider the coupled term as a whole, which is different from the recent results. When applying eigenvalue inequalities to discuss the coupled term, our method has less error. In terms of the properties of special matrices and eigenvalue inequalities, we propose several upper and lower matrix bounds for the solution of DCARE. Further, we discuss the iterative algorithms for the solution of the DCARE. In the fixed point iterative algorithms, the scope of Lipschitz factor is wider than the recent results. Finally, we offer corresponding numerical examples to illustrate the effectiveness of the derived results.     

Discrete generalized algebraic Riccati equations and polynomial matrix factorization

This paper presents an algorithm for solving discrete generalized algebraic Riccati equations with the help of an orthogonal projector. A generalization of the procedure of forming and correcting the orthogonal projector is considered and also that of correcting the proper solution by the Newton-Raphson scheme. The possibility to use the discrete generalized Riccati equations for polynomial matrix factorization with respect to the unit circle is demonstrated. A numerical example is given.     

The limiting form of the continuous algebraic Riccati equation

In this paper solution-preserving transformations of algebraic Riccati equations are examined. As an illustrative example we deal with the limiting cost of the linear-quadratic cheap control problem.     

On upper bounds for the unified algebraic Riccati equation

In recent years, several eigenvalues, norms and determinants bounds have been investigated separately for the solutions of continuous and discrete Riccati equations. In this paper, an upper bound for solution of the unified Riccati equation is presented. In the limiting cases, the result reduces to a new upper bound for the solution of continuous and discrete Riccati equation.     

Closed-form solution for a class of continuous-time algebraic Riccati equations

In the present paper we obtain a closed-form solution for a class of continuous-time algebraic Riccati equations (AREs) with vanishing state weight. The ARE in such a class solves a minimum energy control problem. The obtained closed-form solution is used to prove a link between two independent fundamental limitation results in control over networks.     

On norm bounds for algebraic Riccati and Lyapunov equations

New lower bounds on the spectral norms of the positive definite solutions to the continuos and discrete algebraic matrix Riccati and Lyapunov equations are derived. These bounds are much easier to compute than previously available bounds and appear to be considerably tighter in many cases.     

Lower solution bounds of the continuous coupled algebraic Riccati matrix equation

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.     

On positive definite solutions to the algebraic Riccati equation

Complete necessary and sufficient conditions for the existence of a positive definite solution to the algebraic Riccati equation are given. It is also shown that when a positive definite solution exists, it is either unique, or else there are uncountably many such solutions.     

Eigenvalue bounds in the Lyapunov and Riccati matrix equations

Two related theorems of Strang [1] are extended to provide upper and lower bounds on the eigenvalues of the Lyapunov and Riccati matrices, given byQ=AB^{H}+BAandR=AB^{H}+BA+2AHA, whereAandHare Hermitian, positive definite, complex matrices. We discuss inversion to obtain eigenvalue bounds on the matrixAfor the usual case in whichQ , R, andHare known.     

Curvature properties of the algebraic Riccati equation

We prove that the solution to the algebraic Ricatti equation (ARE) is concave with respect to a nonnegative-definite symmetric state weighting matrix Q when the input weighting matrix R = R^T > 0. We also prove that the solution to the ARE is concave with respect to a positive-definite diagonal input weighting matrix R when Q = Q^T ≥ 0.     

New Upper and Lower Eigenvalue Bounds for the Solution of the Continuous Algebraic Riccati Equation

In this paper, applying majorization inequalities, new upper and lower bounds for summation of eigenvalues (including the trace) of the solution of the continuous algebraic Riccati equation are presented. Corresponding numerical examples illustrate that our new bounds extend some of the recent results.     

A matrix inequality associated with bounds on solutions of algebraic Riccati and Lyapunov equations

A new proof is presented for the inequality,tr (XY) leq parallel X parallel_{2} cdot tr Y. This argument is valid under the condition thatYbe real symmetric nonnegative definite;Xmay be any square matrix.