I have a confusion about whether linear dynamical systems $\dot{x}=Ax(t)+Bu(t)$ are considered autonomous ODEs or nonautonomous. My confusion comes from the input term $Bu(t)$.
The main thing that I want to do is to use this dissertation to calculate the global discetization error of linear dynamical systems. This dissertation considers the general nonautonomous ODEs of the form $\dot{x}=f(x,t)$ (on the page 7) with a given initial condition.
The only mentioned requirement of $f(x,t)$ is that f must be Lipschitz continuous on the domain of $x$ for all t.
Any guidance is greatly appreciated.
Your right hand side depends both on $x$ and $t$, so your system is not autonomous. In general, autonomous means that the evolution from a given state $x$ only depends on that state, but in your case it also depends on the value of $u$ at the given $t$.