Disturbance attenuation problem for discrete-time stochastic systems. The periodic case

An input-output linear time-varying discrete system with state dependent noise and periodic coefficients is considered. Firstly, we define the input-output operator of a such discrete time system and prove that, if the norm of this input-output operator is less then 7 then a corresponding parametrized by 7 Riccati equation has a unique periodic stabilizing solution. An iterative procedure to compute the stabilizing solution of this parametrized Riccati equation is given. Secondly, we prove that if a stabilizing and attenuating feedback exists then a game-theoretic Riccati equation has a unique periodic stabilizing positive semidefinite solution and if such solution exists it allows an explicit construction of a stabilizing and disturbance attenuating state feedback.