Digital control stability for more than impulse response stability?
There might be a problem with your simulation, because for an oscillating input signal (with constant amplitude), the output is also an oscillating signal with constant amplitude, as long as the system is stable, i.e., as long as \$0&lt;k\beta &lt;2\$ is satisfied.
For the given oscillating input signal, the output is given by
$$y[n]=\frac{\beta x_0}{2-k\beta}(-1)^n,\qquad 0&lt;k\beta&lt;2\tag{1}$$
Of course, if the input is switched on at some finite point in time, there will be transients before the output approaches \$(1)\$, but these transients will decay as long as the stability condition \$0&lt;k\beta &lt;2\$ is satisfied.
For values of \$k\beta\$ close to \$2\$ you might be observing the transient, which will increase in amplitude for quite a while, but the output will finally settle at its maximum amplitude \$\beta/(2-k\beta)\$, which can be quite large for values of \$k\beta\$ close to \$2\$.
The figure below shows the output for an oscillating input signal (\$x_0=1\$) starting at \$n=0\$, and for two different values of \$k\beta\$: