Peaking and stabilization

The possibility of globally stabilizing, by means of a smooth state feedback, systems obtained by cascading a linear controllable system and a general nonlinear system is studied. In general, stabilizing the linear part with very negative eigenvalues does not suffice to stabilize the whole system, because of the peaking phenomenon: for a linear controllable system it is always possible to choose a feedback that will have poles with a very negative real part, but in general the resulting trajectories go far away before they go to zero. This poses an obstruction to the stabilization of globally minimum phase systems that can be compensated for if the derivatives of the nonlinear part have sufficiently slow growth. Theorems giving sufficient conditions for stabilizability in terms of such growth conditions are proved.<>