Robust Stabilization of linear stochastic differential models with additive and multiplicative diffusion via attractive ellipsoid techniques

Linear controlled stochastic differential equations (LCSDE) subject to both multiplicative and additive stochastic noises are considered. We study a robust “practical” stabilization for this class of LCSDE meaning that almost all trajectories of this stochastic model converges in a “mean-square sense” to a bounded zone located in an ellipsoidal set. Also, we present a result related to convergence in probability one sense to a zero zone. The considered stabilizing feedback is supposed to be linear. This problem is shown to be converted into the corresponding attractive averaged ellipsoid “minimization” under some constraints of BMI´s (Bilinear Matrix Inequalities) type. The application of an adequate coordinate changing transforms these BMI´s into a set of LMI´s (Linear Matrix Inequalities) that permits to use directly the standard MATLAB - toolbox. A numerical example is used to illustrate the effectiveness of this methodology.