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The system,

enter image description here

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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    $\begingroup$ Welcome to MathSE! Here's a handy guide for how to mark up math nicely on this site, rather than supplying links to images. You are more likely to get a good answer to your question if you follow a few guidelines. In particular, what have you tried so far (or what formulas have your tried to use ), and just where are you stuck? This is not a homework-answering site: we want to see that you have put some thought into the problem. $\endgroup$ – Frentos Nov 4 '17 at 0:51
  • $\begingroup$ I saw a talk (given about 16 years ago) about something called chaos control. The idea there was to give the heart just a little electrical tap at just the right time and it would set the state of the heart rotating about a different strange attractor (from tachycardia back to a sinus rhythm) . If you look up articles using the keywords "chaos control" and "strange attractor" you might get somewhere. I think this article might be the research I saw academic.oup.com/cardiovascres/article/40/2/257/353759/… $\endgroup$ – David Elm Nov 4 '17 at 1:01
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    $\begingroup$ 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 $u$. $\endgroup$ – RTJ Nov 4 '17 at 19:10
  • $\begingroup$ @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. $\endgroup$ – MrYouMath Nov 6 '17 at 9:48
  • $\begingroup$ @MrYouMath I am not sure what you are asking, do you mean a linear system and a nonlinear feedback? $\endgroup$ – RTJ Nov 6 '17 at 13:07

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