Abstract
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. MSC: Primary 15A24; 65F35; secondary 47H10, 47H14.
Highlights
1 Introduction In this paper we consider a general class of continuous-time stochastic algebraic Riccati equations
7 Conclusion While doing numerical computation, it is important in practice to have an accurate method for estimating the relative error and the condition number of the given problems
We focus on providing a tight perturbation bound of the stabilizing solution to stochastic algebraic Riccati equations (SARE) ( a)-( b) under small changes in the coefficient matrices
Summary
The matrix X is called a stabilizing solution for R if the spectrum of the associated operator Lc with respect to X defined by A necessary and sufficient condition for the existence of the stabilizing solution to a more general SARE is derived in Theorem . By using Brouwer fixed point theorem, we obtain a perturbation bound for the stabilizing solution of SARE ( a)-( b) in Section .
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