How to prove that if $(A,B)$ is controllable, then $(A,\bar{B})$ is also controllable, $\bar{B}\bar{B}^T=BB^T+Q$ and $Q$ positive semidefinite I want to prove that if $(A,B)$ is controllable, then for any $\bar{B}$ s.t. $\bar{B}\bar{B}^T=BB^T+Q$, where Q is positive semidefinite, $(A,\bar{B})$ is also controllable. 
I figured I can prove it with the controllability gramian: We know $W(t_0,t_1)=\int_{t_0}^{t_1}\phi(t_1,s)BB^T\phi(t_1,s)^Tds$ for $(A,B)$ will have full rank by controllability. $(A,\bar{B})$ will have $\bar{W}(t_0,t_1)=\int_{t_0}^{t_1}\phi(t_1,s)BB^T\phi(t_1,s)^Tds + \int_{t_0}^{t_1}\phi(t_1,s)Q\phi(t_1,s)^Tds$. How do I prove $\bar{W}(t_0,t_1)$ is full rank?
 A: You are on the right track. Namely, the controllability Gramians are given by
$$
W(t_0,t_1) = \int_{t_0}^{t_1} \phi(t_1,s)\,B\,B^\top \phi(t_1,s)^\top ds, \tag{1}
$$
$$
\bar{W}(t_0,t_1) = \int_{t_0}^{t_1} \phi(t_1,s)\,\bar{B}\,\bar{B}^\top \phi(t_1,s)^\top ds, \tag{2}
$$
with $\bar{B}\,\bar{B}^\top = B\,B^\top + Q$ and $Q$ positive semi-definite. So $(2)$ can also be written in terms of $(1)$
$$
\bar{W}(t_0,t_1) = W(t_0,t_1) + \int_{t_0}^{t_1} \phi(t_1,s)\,Q\,\phi(t_1,s)^\top ds. \tag{3}
$$
Since $Q$ is positive semi-definite the last integral term in $(3)$ should also be positive semi-definite if $t_1 \geq t_0$. So for that same time interval $\bar{W}(t_0,t_1)$ would always be lower bounded by $W(t_0,t_1)$. 
A linear (possibly time-varying) system $(A(t),B(t))$ is state controllable if and only if the controllability Gramian is full rank for all $t_1 > t_0$ [1].
State controllable means that there always exists a control input sequence $u(t)$ which drives the system from any state $x(t_0) = x_0$ to any given state $x_1$ at $t_1 > t_0$. There are also other definitions related controllability, such as 0-controllability and reachability, which in many cases are all equivalent. However, there might be some cases one does not imply the other, but I believe state controllability does always imply the others. To show that a full rank controllability Gramian (which also implies positive definite, check this yourself) implies that the system is state controllable one can use that the general solution over time of such a system can be expressed as
$$
x(t) = \phi(t,t_0)\,x(t_0) + \int_{t_0}^t \phi(t,\tau)\,B(\tau)\,u(\tau)\,d\tau, \tag{4}
$$
with $\phi(t_1,t_0)$ the state transition matrix (which only depends on $A(t)$). The following proposed control input should drive the state of the system to $x_1$ at $t=t_1$
$$
u(t) = B(t)^\top\phi(t_1,t)^\top W_c^{-1}(t_0,t_1) \left(x_1 - \phi(t_1,t_0)\,x_0\right), \tag{5}
$$
with 
$$
W_c(t_0,t_1) = \int_{t_0}^{t_1} \phi(t_1,s)\,B(s)\,B(s)^\top\phi(t_1,s)^\top ds. \tag{6}
$$
Namely, evaluating the integral from $(4)$ using $(5)$ at $t = t_1$ and $(6)$ gives
\begin{align}
\int_{t_0}^{t_1} \phi(t_1,\tau)\,B(\tau)\,u(\tau)\,d\tau &= \int_{t_0}^{t_1} \phi(t_1,\tau)\,B(\tau)\,B(\tau)^\top\phi(t_1,\tau)^\top W_c^{-1}(t_0,t_1) \left(x_1 - \phi(t_1,t_0)\,x_0\right)\,d\tau, \\
&= \int_{t_0}^{t_1} \phi(t_1,\tau)\,B(\tau)\,B(\tau)^\top\phi(t_1,\tau)^\top d\tau\, W_c^{-1}(t_0,t_1) \left(x_1 - \phi(t_1,t_0)\,x_0\right), \\
&= W_c(t_0,t_1)\,W_c^{-1}(t_0,t_1) \left(x_1 - \phi(t_1,t_0)\,x_0\right), \\
&= x_1 - \phi(t_1,t_0)\,x_0.
\end{align}
Plugging this into $(4)$ gives
$$
x(t_1) = \phi(t_1,t_0)\,x(t_0) + x_1 - \phi(t_1,t_0)\,x_0 = x_1,
$$
and thus completing the proof that $(A(t),B(t))$ is state controllable if the controllability Gramian is full rank, such that the inverse of $W_c(t_0,t_1)$ exists.
Now since $(A,B)$ is said to be controllable it must hold that $W(t_0,t_1)$ is positive definite for $t_1 > t_0$, so $\bar{W}(t_0,t_1)$ should be positive definite as well. So since $\bar{W}(t_0,t_1)$ should be positive definite for $t_1 > t_0$ it follows that $(A,\bar{B})$ is also controllable.
[1]: Skogestad, S., & Postlethwaite, I. (2007). Multivariable feedback control: analysis and design (pp. 128). Wiley.
