Guaranteed-cost stabilization of non-linear stochastic systems

The control of uncertain non-linear discrete-time systems having stochastic cone-bounded non-linearities is considered. First, a quadratic performance bound and a guaranteed-cost optimal state feedback controller are derived. Then, an auxiliary system is introduced. It is shown that the quadratic optimal control for this auxiliary system is the same as the guaranteed-cost control for the original system, and, therefore, the existence of the infinite-horizon guaranteed-cost controller can be based on the stabilizability and observability properties of the auxiliary system. Finally, the stochastic boundedness of the controlled uncertain system is proved based on the properties of the auxiliary system.