Stochastic linear systems subjected both to Markov jumps and to multiplicative white noise are considered. In order to stabilize such type of stochastic systems, the so-called set of generalized discrete-time algebraic Riccati equations has to be solved. The LMI approach for computing the stabilizing symmetric solution (which is in fact the equilibrium point) of this system is studied. We construct a new modification of the standard LMI approach, and we show how to apply the new modification. Computer realizations of all modifications are compared. Numerical experiments are given where the LMI modifications are numerically compared. Based on the experiments the main conclusion is that the new LMI modification is faster than the standard LMI approach. 1. Introduction In this paper we investigate general stochastic algebraic Riccati equations which are related to LQ control models for stochastic linear systems with multiplicative white noise and markovian jumping. We consider a stochastic system described by where is a -dimensional standard Brownian motion with and , defined by a filtered probability space . In addition, , is a right continuous homogeneous Markov chain with the state space the set and the probability transition matrix , , and with , , and if . It is assumed that the and are independent stochastic processes and for all . The state vector is a real vector, denotes the vector of control variables, and is the regulated output vector with components. The matrix coefficients , , , are constant matrices of appropriate dimensions with real elements. The stochastic systems with multiplicative white noise naturally arise in control problems of linear uncertain systems with stochastic uncertainty. It is important for applications to find a stabilizing controller for the above stochastic system (for more details see [1, 2]). For this purpose it is enough to compute the stabilizing solution to the following stochastic generalized Riccati algebraic equations: The concepts of stabilizing solution of the Riccati-type equation (2) and of stabilizability for the triple , where as usual , are defined in a standard way (see [1]). Applying Theorem 4.9 of Dragan and Morozan [2] we deduce that system (2) has a unique stabilizing solution with . The control stabilizes system (1). An e1ective iterative convergent algorithm to compute these stabilizing solutions is presented in [3]. Let us consider the special case of (1) where . The stochastic linear quadratic model studied by Yao et al. in [4] is obtained. In this model the special functional is minimized (see
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