Given that the LTV system $\dot x = A\left( t \right)x\left( t \right) + B\left( t \right)u\left( t \right)$ is controllable, how can I show that $\dot x = \left( {A\left( t \right) + B\left( t \right)K\left( t \right)} \right)x\left( t \right) + B\left( t \right)k\left( t \right)$ is controllable also?
my attempt: LTV system is controllable if the controllability Gramian has rank $n$ for all $t$. Obviously I have to use the fact that the nominal system is controllable but I do not seem to get any idea how to start. Any tips are appreciated!
Suppose $\dot{x} = A(t)x+B(t)u$ is controllable on some interval $[t_0,t_1]$. I am interpreting this as meaning for any two states $x_0,x_1$ there is a control $u$ such that with $x(t_0) = x_0$ we end up with $x(t_1) = x_1$.
Now consider the system $\dot{y} = (A(t)+K(t)B(t))y+B(t)k$ with initial condition $y(t_0) = x_0$. Choose the control $k = u-K(t) y$, then the resulting ODE is $\dot{y} = A(t)y+B(t)u$ and hence $y(t) = x(t)$ for $t \in [t_0,t_1]$. In particular, $y(t_1) = x_1$. Hence this system is controllable.