Global stabilization of fixed points using predictive control

We analyze the global stability properties of some methods of predictive control. We particularly focus on the optimal control function introduced by de Sousa Vieira and Lichtenberg [Phys. Rev. E 54, 1200 (1996)]. We rigorously prove that it is possible to use this method for the global stabilization of a discrete system x(n+1)=f(x(n)) into a positive equilibrium for a class of maps commonly used in population dynamics. Moreover, the controlled system is globally stable for all values of the control parameter for which it is locally asymptotically stable. Our study highlights the difficulty of obtaining global stability results for other methods of predictive control, where higher iterations of f are used in the control scheme.