# Can a system that cannot be stabilized with full state feedback be still stabilizable?

The system,

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cannot be stabilized with full state feedback, and the determinant of its controllability matrix is 0. Hence it is not controllable either.

But can it still be stabilizable? - Maybe using something else than full state feedback.

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• The answer is no. Note that for the second state you have $\dot{x}_2=2x_2$ with solution that grows exponentially $x_2(t)=e^{2t}x_2(0)$ that cannot be changed by any selection of the control $u$. – RTJ Nov 4 '17 at 19:10
• @CTNT: Can you give more details on your answer. For linear feedback your answer is obvious but what about nonlinear feedback? I was thinking about a nonlinear $u$ that drives $x_1$ and $x_2$ to zero. But I was not able to come up with an example. – MrYouMath Nov 6 '17 at 9:48
• @MrYouMath I am not sure what you are asking, do you mean a linear system and a nonlinear feedback? – RTJ Nov 6 '17 at 13:07