DT State-Space State Feedback Control Suppose we are given the following discrete-time system:
\begin{equation}
\begin{bmatrix}
x_1(k+1)\\
x_2(k+1)
\end{bmatrix} = \begin{bmatrix}
0 && 1\\
-2 && 0
\end{bmatrix}\begin{bmatrix}
x_1(k)\\
x_2(k)
\end{bmatrix} + \begin{bmatrix}
0\\
1
\end{bmatrix}u(k)
\end{equation}
\begin{equation}
y_1 = \begin{bmatrix}
1 && 1
\end{bmatrix}\begin{bmatrix}
x_1\\
x_2
\end{bmatrix}  
\end{equation}
For this system we can construct a state feedback in the form
$$u(k) = v(k) + \begin{bmatrix}
k_1 && k_2
\end{bmatrix}\begin{bmatrix}
x_1(k)\\
x_2(k)
\end{bmatrix}$$
for which the system has a double pole at some $\beta$. I can design such a state feedback.
Now, suppose this desgined feedback can be active for one time step and the systems returns to its original form (no feedback) until the the next re-activation of the feedback.  This re-activation takes 2 time steps. I want to find the value $\beta$ to maintain the stability. I'm confused at this question since the stability is considered as $k$ goes $\infty$.
 A: Based on the comment, this is a periodic system that can be written as
$$x(3(k+1))=A^2(A+BK)x(3k).$$
Asymptotic stability is ensured provided that the eigenvalues of $A^2(A+BK)$ are inside the unit disc. A way to ensure that is through the consideration of a quadratic Lyapunov function of the form $V(x)=x^TPx$ where $P$ symmetric positive definite (i.e. $P\succ0$). This yields the following Linear Matrix Inequality
$$\begin{bmatrix}
-Q & A^3Q+A^2BU\\(A^3Q+A^2BU)^T & -Q
\end{bmatrix}\prec0$$
where $U$ and $Q\succ0)$ are matrices to be computed and the inequality sign means that the above expression is negative definite. A suitable gain $K$ can be constructed using the expression $K=UQ^{-1}$.
This can be slightly modified by considering a scalar $\beta\in(0,1)$ and solving for the condition
$$\begin{bmatrix}
-\beta^2 Q & A^3Q+A^2BU\\(A^3Q+A^2BU)^T & -Q
\end{bmatrix}\prec0,$$
which is equivalent to saying that the spectral radius of $A^2(A+BK)$ is less than $\beta$.
A more direct approach (see Kwin's comment) is to just design a state-feedback controller for the system $(A^3,A^2B)$. It is interesting to note that this system may not be controllable even if $(A,B)$ is. A simple example is the integrator $\dot{x}=u$.
