Global feedback stabilization of discrete-time bilinear systems

This paper studies the stabilization of the discrete-time bilinear system z(k+1)=Az(k)+u kBz(k), where A and B are bounded linear operators on a pre-Hilbert space H, by means of a new nonlinear state feedback control which is uniformly bounded. First, sufficient conditions for uniform exponential stabilization are given, and the decay rate of the state is estimated. Then global weak stabilization is investigated under less stringent assumptions. In the case of the finite dimension, easily verified conditions for uniform stabilization are given. Moreover, a decomposition method allows additional stabilization results. Finally, illustrative examples are provided.