# Show that if a linear dynamical equation is controllable at $t_0$, then it is controllable at any $t<t_0$.

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?

• Do you mean a linear time invariant system, because general linear systems also include linear time variant systems? – Kwin van der Veen Nov 22 '18 at 0:20
• yeah linear time invariant system – Ali G Nov 22 '18 at 3:54
• Without loss of gbenerality, you can show if the terminal state $x_1=0$ at time $t_0$, then $x(t)=0$ for all $t>t_0$ . – Gustave Nov 23 '18 at 14:39
• What have you done? Perhaps it would be useful to find the solution of the system for $x$. Can you write down this formula? – SZN Nov 27 '18 at 3:37

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.