Perturbation analysis of the stochastic algebraic Riccati equation

Chun Yueh Chiang; Hung Yuan Fan; Matthew M. Lin; Hsin An Chen

doi:10.1186/1029-242X-2013-580

Perturbation analysis of the stochastic algebraic Riccati equation

Chun Yueh Chiang, Hung Yuan Fan, Matthew M. Lin^*, Hsin An Chen

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

4 Citations (Scopus)

Abstract

In this paper we study a general class of stochastic algebraic Riccati equations (SARE) arising from the indefinite linear quadratic control and stochastic H_∞ problems. Using the Brouwer fixed point theorem, we provide sufficient conditions for the existence of a stabilizing solution of the perturbed SARE. We obtain a theoretical perturbation bound for measuring accurately the relative error in the exact solution of the SARE. Moreover, we slightly modify the condition theory developed by Rice and provide explicit expressions of the condition number with respect to the stabilizing solution of the SARE. A numerical example is applied to illustrate the sharpness of the perturbation bound and its correspondence with the condition number.

Original language	English
Article number	580
Journal	Journal of Inequalities and Applications
Volume	2013
Issue number	1
DOIs	https://doi.org/10.1186/1029-242X-2013-580
Publication status	Published - 2013

Keywords

Brouwer fixed-point theorem
Condition number
Perturbation bound
Stochastic algebraic riccati equations

ASJC Scopus subject areas

Analysis
Discrete Mathematics and Combinatorics
Applied Mathematics

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10.1186/1029-242X-2013-580

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title = "Perturbation analysis of the stochastic algebraic Riccati equation",

abstract = "In this paper we study a general class of stochastic algebraic Riccati equations (SARE) arising from the indefinite linear quadratic control and stochastic H∞ problems. Using the Brouwer fixed point theorem, we provide sufficient conditions for the existence of a stabilizing solution of the perturbed SARE. We obtain a theoretical perturbation bound for measuring accurately the relative error in the exact solution of the SARE. Moreover, we slightly modify the condition theory developed by Rice and provide explicit expressions of the condition number with respect to the stabilizing solution of the SARE. A numerical example is applied to illustrate the sharpness of the perturbation bound and its correspondence with the condition number.",

keywords = "Brouwer fixed-point theorem, Condition number, Perturbation bound, Stochastic algebraic riccati equations",

author = "Chiang, {Chun Yueh} and Fan, {Hung Yuan} and Lin, {Matthew M.} and Chen, {Hsin An}",

note = "Funding Information: The authors wish to thank the editor and two anonymous referees for many interesting and valuable suggestions on the manuscript. This research work is partially supported by the National Science Council and the National Center for Theoretical Sciences in Taiwan. The first author was supported by the National Science Council of Taiwan under Grant NSC 102-2115-M-150-002. The second author was supported by the National Science Council of Taiwan under Grant NSC 102-2115-M-003-009. The third author was supported by the National Science Council of Taiwan under Grant NSC 101-2115-M-194-007-MY3. Publisher Copyright: {\textcopyright} 2013 Chiang et al.",

year = "2013",

doi = "10.1186/1029-242X-2013-580",

language = "English",

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AU - Chiang, Chun Yueh

AU - Fan, Hung Yuan

AU - Lin, Matthew M.

AU - Chen, Hsin An

N1 - Funding Information: The authors wish to thank the editor and two anonymous referees for many interesting and valuable suggestions on the manuscript. This research work is partially supported by the National Science Council and the National Center for Theoretical Sciences in Taiwan. The first author was supported by the National Science Council of Taiwan under Grant NSC 102-2115-M-150-002. The second author was supported by the National Science Council of Taiwan under Grant NSC 102-2115-M-003-009. The third author was supported by the National Science Council of Taiwan under Grant NSC 101-2115-M-194-007-MY3. Publisher Copyright: © 2013 Chiang et al.

PY - 2013

Y1 - 2013

N2 - In this paper we study a general class of stochastic algebraic Riccati equations (SARE) arising from the indefinite linear quadratic control and stochastic H∞ problems. Using the Brouwer fixed point theorem, we provide sufficient conditions for the existence of a stabilizing solution of the perturbed SARE. We obtain a theoretical perturbation bound for measuring accurately the relative error in the exact solution of the SARE. Moreover, we slightly modify the condition theory developed by Rice and provide explicit expressions of the condition number with respect to the stabilizing solution of the SARE. A numerical example is applied to illustrate the sharpness of the perturbation bound and its correspondence with the condition number.

AB - In this paper we study a general class of stochastic algebraic Riccati equations (SARE) arising from the indefinite linear quadratic control and stochastic H∞ problems. Using the Brouwer fixed point theorem, we provide sufficient conditions for the existence of a stabilizing solution of the perturbed SARE. We obtain a theoretical perturbation bound for measuring accurately the relative error in the exact solution of the SARE. Moreover, we slightly modify the condition theory developed by Rice and provide explicit expressions of the condition number with respect to the stabilizing solution of the SARE. A numerical example is applied to illustrate the sharpness of the perturbation bound and its correspondence with the condition number.

KW - Brouwer fixed-point theorem

KW - Condition number

KW - Perturbation bound

KW - Stochastic algebraic riccati equations

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Perturbation analysis of the stochastic algebraic Riccati equation

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Keywords

ASJC Scopus subject areas

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