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Consider a $n$-dimentional $p$-input equation: $$\dot{x}=Ax+Bu$$ where $A$ and $B$ are constant $n\times n$ and $n\times p$ real matrices.

By definition, the latter state equation is said to be controllable if for any initial state $x(0)=x_0$ and any final state $x_1$, there exists an input that transfers $x_0$ to $x_1$ in a finite time.

Then, how can I show that if a linear dynamical equation is controllable at $t_0$ then it is controllable at any $t<t_0$?

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  • $\begingroup$ Do you mean a linear time invariant system, because general linear systems also include linear time variant systems? $\endgroup$ – Kwin van der Veen Nov 22 '18 at 0:20
  • $\begingroup$ yeah linear time invariant system $\endgroup$ – Ali G Nov 22 '18 at 3:54
  • $\begingroup$ Without loss of gbenerality, you can show if the terminal state $x_1=0$ at time $t_0$, then $x(t)=0$ for all $t>t_0$ . $\endgroup$ – Gustave Nov 23 '18 at 14:39
  • $\begingroup$ What have you done? Perhaps it would be useful to find the solution of the system for $x$. Can you write down this formula? $\endgroup$ – SZN Nov 27 '18 at 3:37
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Recall that $\dot{x} = Ax + Bu$, then $(A,B)$ is controllable with control law (with minimum energy) given by

$$u(t) = -B^{T}e^{A^{T}(t_1-t)}W_c^{-1}(t_1)(e^{At_1}x_0 -x_1), $$ where $$W_c(t) = \int_0^te^{A\tau}BB^{T}e^{A^{T}\tau}d\tau.$$

For $W_c$ nonsingular, then equivalence is $(A,B)$ is controllable.

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