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In general, when we talk about controllability, we talk about proving the existence of a control input that transfers the state to a desired state at a desired time $T$. However, when we talk about stability, we prove that the solution tends to zero when time tends to infinity.

Is there any relation between the stability (exponential, polynomial, strong) and the controllability (approximate or exact, null) of PDEs?

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These two problems are fundamentally different, because their data are. As far as linear systems are concerned, controlability is a property for an ODE $\dot x=Ax+Bu$, where $u$ is the control, while stability is a property of an ODE $\dot x=Ax$. On the one hand, the data is a pair $(A,B)$. On the other hand, the data is a square matrix $A$. The first system is controllable if the range of $(B,AB,A^2B,\ldots)$ is full, while the second is stable if the spectrum of $A$ has negative real part. Definitively independent properties.

Similar considerations are valid for nonlinear systems.

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I found the answer of my question in the following book, https://books.google.dz/books?id=wRdiJqs8F5QC&printsec=frontcover&dq=mathematical+control+theory&hl=fr&sa=X&ved=0ahUKEwihv6mkwNXWAhXBVhoKHVD-C64Q6AEILDAB

the null controllability is the only type of control who applies stability, there is any relation between other types and stability.

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  • $\begingroup$ If this answers your question, you should mark this as the accepted answer by clicking the green checkmark to the left. $\endgroup$
    – j.c.
    Commented Aug 30, 2018 at 15:10
  • $\begingroup$ @Gustave How null controllability implies stability? $\endgroup$
    – Migalobe
    Commented Jun 28, 2020 at 1:46

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