Difference between an "controlled ODE" and a "non-homogeneous ODE" In control theory one consider so called "controlled odes" i.e $X^{'}=AX+Bu$ where $u$ is the controll or input and $X$ is the "state". This in turn is describing a controlled dynamical system.
In the theory of ODEs we have homogeneous and non-homogeneous ODEs. The non-homogeneous often taking the form say, $y^{'}+Ay=u$ for some given function $u$.
Is there any essential difference between these to notions? I cannot think of any.
 A: In the the non-homogeneous ODE case $u$ would be a function of time. The response of the system will be a sum of the transient and steady-state response of the system. The transient will only depend of the initial conditions of $x$ and $A$, so a homogeneous solution. This transient response of the system will die out eventually if the system is stable. For a linear time invariant state space model this is the case when all the eigenvalues of $A$, also known as the poles, have negative real part. The steady-state will depend on $u$, $A$ and $B$.
In the controlled ODE case $u$ could be a function of both time and the state $x$. The fact that $u$ can be a function of $x$ means that one will be able to alter the dynamics of the system, so change the location of the poles. This, if the pair $(A,B)$ is controllable/stabilizable, allows you to turn an unstable system into a stable one.
So to summarize, the non-homogeneous ODE only allows you to change the steady-state of the system (which only "exists" if the system is stable by itself). While the controlled ODE allows you to change both the transient and the steady-state.
