Newtonʹs method for a rational matrix equation occurring in stochastic control

We study a general class of rational matrix equations, which contains the continuous (CARE) and discrete (DARE) algebraic Riccati equations as special cases. Equations of this type were encountered in [SIAM J. Control and Optimization 36 (1998) 1504–1538; Stochastics and Stochastics Reports, 65 (1999) 255–297], where H∞-type problems of disturbance attenuation for stochastic linear systems were studied. We develop a unifying framework for the analysis of these equations based on the theory of (resolvent) positive operators and show that they can be solved by Newtonʹs method starting at an arbitrary stabilizing matrix.