Ultimate boundedness and robust stabilization of bilinear discrete-time systems

In this paper nonlinear time-varying and bilinear discrete-time systems with additive bounded disturbances are considered. First, conditions guaranteeing uniform ultimate boundedness for time-varying nonlinear systems are established. Then, algebraic conditions ensuring the existence of polyhedral Lyapunov functions, uniform boundedness and positive invariance for closed-loop bilinear systems are obtained. Finally, these results are applied to various robust stabilization problems for bilinear systems subject to persistent additive disturbances. It is shown that these problems can be reduced to a single or a series of linear programming problems.