If a pole lies on the imaginary axis, the system is:

• A. Stable
• B. Unstable
• C. Marginally stable
• D. Overdamped

Answer: (C)

When a pole lies on the imaginary axis in the context of control systems and stability analysis, it indicates that the system's response neither decays to zero nor grows unbounded over time. Instead, the system oscillates indefinitely at a constant amplitude, typically characterized by sustained oscillations without exponential decay. This behavior is indicative of marginal stability.

In technical terms, a pole located exactly at the imaginary axis means that on the complex plane, the real part of the pole is zero, which leads to oscillatory modes that do not settle down. Therefore, the system does not meet the criteria for either stability (where all poles have negative real parts and the output decays to zero) or instability (where at least one pole has a positive real part, causing the output to grow unbounded).

The concepts of overdamping, stability, and instability refer to different configurations of poles, where overdamped systems typically have all poles in the left half-plane and lead to a slow return to equilibrium without oscillations. In contrast, the condition described by a pole on the imaginary axis is specifically associated with marginal stability, forming the correct choice.