In a finite (continuous) time, LQR problem, which is used for modeling a real world problem, Does the number of state variables (m) and the number of control variables (n) has any conditions? For example can we have 5 state variables and 25 control variables?
Generally, I didn't see any text book regarding this question.
In order to avoid confusion I will state how I define the state space model
$$ \dot{x}(t) = A\,x(t) + B\,u(t) $$
and the cost function
$$ J = \int_0^T x^\top\!(t)\,Q\,x(t) + u^\top\!(t)\,R\,u(t)\,dt + x^\top\!(T)\,P_T\,x(T). $$
For the infinite horizon LQR problem it is required that $(A,B)$ is stabilizable, $R \succ 0$ and $(A,Q)$ has no unobservable modes on the imaginary axis (this condition changes slightly when one also includes a cross terms $x^\top N\,u$ in the cost function). However, the finite time LQR problem is less strict, namely only $R \succ 0$ (and $P_T \succeq 0$). This is because the other constraints are there to ensure that the limit of time to infinity is well defined, but this is not required for finite time (since linear systems don't blow up to infinity in finite time). The constraint $R \succ 0$ is still required to prevent trying to apply infinite control signals.
So the size of the state and number of inputs is not important. The only issue might be that numerically solving the Riccati differential equation might become more computationally expensive to solve for larger values.