Stabilization in probability of stochastic nonlinear quadratic systems with guaranteed cost control

We present new results regarding the stability properties of a stochastic nonlinear quadratic system (NLQS). The paper extends to the stochastic context a previous work concerning the domain of attraction of the zero equilibrium point of a NLQ. In this context, we use the concept of (Ω,α)-stability in probability and we achieve sufficient stability condition by exploiting the usual approach based on quadratic Lyapunov. This approach allows us to solve also the stabilization problem obtaining a procedure to design a state feedback control law which guarantees a region of attraction with a certain level of risk. The proposed designed procedure requires the solution of an optimization problem in the form of linear matrix inequalities, which allows us to estimate an upper bound for the quadratic performance functional. Two examples based on biological phenomena illustrate the effectiveness of the developed approach.