I've been researching a problem in non-linear control and I've come up
with a problem that seems rather easy but nevertheless have not been
able to tackle it. Any help is appreciated. Here goes:

Let x'=f(x,u) be a non linear system. The controls uEU (where U is a bounded subset of R^m, i.e. the controls are bounded. "E" denotes the "belongs to" operator). Suppose you have a stabilizing feedback law u(x) such that x->0 (with f(0,0)=0 being an equilibrium point). Now consider that you change the bounds of the control space through a transform F(U)=U' such that 0EU' and the intersection of U and U' is not empty. And the question is: Does the control u(x) still stabilizes the system? Or, more strictly speaking, does the feedback law s(u(x)), where s is a saturation function that cuts off u(x) when it exceeds the bounds of U', stabilizes the system?

Let x'=f(x,u) be a non linear system. The controls uEU (where U is a bounded subset of R^m, i.e. the controls are bounded. "E" denotes the "belongs to" operator). Suppose you have a stabilizing feedback law u(x) such that x->0 (with f(0,0)=0 being an equilibrium point). Now consider that you change the bounds of the control space through a transform F(U)=U' such that 0EU' and the intersection of U and U' is not empty. And the question is: Does the control u(x) still stabilizes the system? Or, more strictly speaking, does the feedback law s(u(x)), where s is a saturation function that cuts off u(x) when it exceeds the bounds of U', stabilizes the system?