Local assignability of Lyapunov exponents of linear discrete-time system

We consider sufficient and necessary conditions for the proportional local assignability of the Lyapunov spectrum of the system
$$ x(m+1)=\left(A(m)+B(m)U(m)\right)x(m), \quad m\in\mathbb Z,\quad x\in\mathbb R^n. $$
The properties of stability of the Lyapunov spectrum and integral separation of linear discrete-time systems are studied, description of the spectral set of a linear system in the case of the full spectrum stability is obtained, the property of uniform complete controllability of a linear system with discrete time is studied, and the properties of the Bebutov shell of a linear discrete-time control system are investigated.