Attainability, Lyapunov Reducibility, and Assignability of Lyapunov Invariants of Linear Discrete-Time Systems

In this article, we consider a linear discrete time-invariant control system with linear time-varying state feedback. Our main goal is to determine what types of asymptotic dynamics are possible for this system. We introduce and investigate certain discrete-time analogs for notions of global attainability, global Lyapunov reducibility, and global assignability of Lyapunov invariants currently known in the continuous-time case. We prove that uniform complete controllability of the open-loop system is necessary for uniform global nonsingular attainability of the closed-loop system and is sufficient if the coefficients of the open-loop system are time invariant. From here, we demonstrate that if a given discrete-time linear time-invariant system is completely controllable, then, by choosing a suitable time-varying feedback, it is possible to yield a dynamic equivalence between the closed-loop system and an arbitrary discrete time-varying linear system, which has a Lyapunov sequence as its coefficient matrix.