To check the BIBO (Bounded Input, Bounded Output) stability of a system with a given system function or difference equation, the general approach is as follows:

For a system described by a linear difference equation like:

$ax(n) = x(n-1)$

The main steps are:

1. Express the transfer function

To analyze BIBO stability, you first need to express the system's transfer function, $H(z)$, which is the Z-transform of the system's impulse response.

Start by reformatting the difference equation:

$ax(n) = x(n-1)$

This can be rewritten as:

$ax(n) - x(n-1) = 0$

Taking the Z-transform of both sides gives us:

$aX(z) - z^{-1}X(z) = 0$

Factor out $X(z)$:

$X(z)(a - z^{-1}) = 0$

Thus, the transfer function $H(z)$ is:

$H(z) = \frac{X(z)}{X(z)} = \frac{1}{a - z^{-1}}$

2. Check the poles of the transfer function

For BIBO stability, all the poles of the transfer function must lie inside the unit circle in the z-plane, meaning the absolute value of each pole must be less than 1.

In this case, the transfer function $H(z)$ doesn't explicitly have poles yet, but we need to consider if any poles exist.

3. Conclusion on BIBO Stability

For a system to be BIBO stable, we need to investigate if any condition exists on $a$ that could lead to poles outside the unit circle. In this case, the system seems degenerate (since the input equals the output in steady state), implying that the system is marginally stable. However, we would generally need more information about the exact nature of $a$ and the setup.

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Would you like more details on this, or do you have any questions?
Here are five follow-up questions for further exploration:

1. How does BIBO stability relate to internal system stability?
2. Can you always derive a transfer function from a difference equation?
3. How does the location of poles affect the system's transient response?
4. What are the practical implications of a system being marginally stable?
5. How do the properties of $a$ influence stability in this particular equation?

Tip: When checking BIBO stability, always remember that all poles of the transfer function must be within the unit circle in the complex plane for the system to be considered stable.