Perturbation analysis and condition numbers of rational Riccati equations

In this paper, we consider the perturbation analyses of the discretetime rational Riccati equations using the normwise, mixed and componentwise analyses, which arises from the stochastic $H_\infty$ problems and the indefinite stochastic linear quadratic control problems. We derive sufficient conditions for the existence of stabilizing solutions of the perturbed rational Riccati equations. Moreover, we obtain the perturbation bounds for the relative errors with respect to the stabilizing solutions of the rational Riccati equations under three kinds of perturbation analyses. Numerical results are presented to illustrate sharper perturbation bounds under the normwise, mixed and componentwise perturbation analyses.

Keywords

rational Riccati equation, perturbation bound, mixed condition number, componentwise perturbation analysis, stochastic optimal control

2010 Mathematics Subject Classification

15A24, 47H10, 65F35, 93C05, 93E20

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This work was supported by (a) Career Development Award of Academia Sinica (Taiwan) grant number 103-CDA-M04 for Phoa and Weng, (b) Ministry of Science and Technology (Taiwan) grant numbers 104-2118-M-001-016-MY2 and 105-2118-M-001-007-MY2 for Phoa, and (c) Ministry of Science and Technology (Taiwan) grant number 103-2811-M-001-166 for Weng.

Received 25 February 2021

Accepted 13 March 2021

Published 6 October 2021