Abstract
Nonsymmetric differential matrix Riccati equations arise in many problems related to science and engineering. This work is focusing on the sensitivity of the solution to perturbations in the matrix coefficients and the initial condition. Two approaches of nonlocal perturbation analysis of the symmetric differential Riccati equation are extended to the nonsymmetric case. Applying the techniques of Fréchet derivatives, Lyapunov majorants and fixed-point principle, two perturbation bounds are derived: the first one is based on the integral form of the solution and the second one considers the equivalent solution to the initial value problem of the associated differential system. The first bound is derived for the nonsymmetric differential Riccati equation in its general form. The perturbation bound based on the sensitivity analysis of the associated linear differential system is formulated for the low-dimensional approximate solution to the large-scale nonsymmetric differential Riccati equation. The two bounds exploit the existing sensitivity estimates for the matrix exponential and are alternative.
Highlights
Introduction and NotationsIn the present paper, we consider the nonsymmetric differential matrix Riccati equation /NDRE/X (t) = −AX(t) − X(t)D + X(t)SX(t) + Q, (1) X(0) = X0.where the solution X(t) is a n × p real matrix and A ∈ Rn×n, D ∈ Rp×p, Q ∈ Rn×p and S ∈ Rp×n are the coefficient matrices and X0 ∈ Rn×p is a given initial value.We assume that the matrix H= D −S −Q A∈ Rp+n×p+n is a nonsingular M-matrix, or an irreducible singular M-matrix. (Recall that a real square matrix A is said M-matrix if A = sI − B with B ≥ 0 and s ≥ r(B), where r(.) denotes the spectral radius
Two computable perturbation bounds are derived using the techniques of Fréchet derivatives, Lyapunov majorants and fixed-point principles, developed in [14]
The second one exploits the statement of the classical Radon’s theory of local equivalence of the solution to the differential matrix Riccati equation to the solution of the initial value problem of the associated differential system. It has the advantage of not being related with the solution of the NDRE and with problems of divergence of the numerical procedure
Summary
We consider the nonsymmetric differential matrix Riccati equation /NDRE/. In [1], it is proved that if H is assumed to be a nonsingular M-matrix, the NDRE (1) has a global solution X(t), provided that the initial value X0 satisfies the condition 0 ≤ X0 ≤ X∗, where for every matrices A, B ∈ Rm×n, we write A ≤ B if aij ≤ bij for all i ∈ {1, . Mathematics 2021, 9, 855 nonlocal perturbation bound to estimate the error of approximation in the solution when solving large-scale nonsymmetric differential Riccati equations by Krylov-type methods. The paper is organized as follows: In Section 2, nonlocal sensitivity analysis of the nonsymmetric differential matrix Riccati Equation (1) is presented. F is the Frobenius norm, A is the transpose of the matrix A ∈ Rm×n, In is the n × n unit matrix, and the symbol := stands for “equal by definition”
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