On the stabilization of nonlinear systems

We extend the applicability of the global
Q
-parametrization method of controller design to a large class of unstable nonlinear plants. The main result is a two-step compensation theorem analogous to that of Zames for unstable linear plants-if
P: L_{e2} \rightarrow L_{e1}
is a nonlinear (possibly unstable) plant and F
0
is any incrementally stable controller such that
P_{1}:=P(I - F_{0}(-P))^{-1}
is incrementally stable, then the class of controllers
F
which yields an f.g. stable closed-loop system in the unity feedback configuration for
P
is globally parametrized by finite gain stable maps
Q: L_{e1} \rightarrow L_{e2}
with
F = F_{0} + Q(I - P_{-1}Q)^{-1}