Final Value Theorem: Identify the correct statement for a stable system output y(t) with Laplace transform Y(s).

Introduction / Context:
The Final Value Theorem (FVT) links steady-state time behavior to a limit in the Laplace domain. It is widely used to compute steady-state error in control systems without performing inverse transforms.

Given Data / Assumptions:

• Output y(t) has Laplace transform Y(s).
• System stability assumptions are critical to FVT validity.
• No poles on the imaginary axis (except possibly a simple pole at the origin in sY(s) is not allowed).

Concept / Approach:

The FVT states: if all poles of sY(s) lie strictly in the left half-plane (no poles at or to the right of jω axis), then lim_{t→∞} y(t) exists and equals lim_{s→0} s Y(s). The condition on sY(s) ensures convergence and rules out oscillatory or divergent steady-state behavior.

Step-by-Step Solution:

Verification / Alternative check:

For a unit-step input through a stable first-order system G(s) = K/(τ s + 1), Y(s) = G(s)/s. Then sY(s) = K/(τ s + 1) and lim_{s→0} sY(s) = K, matching the known steady-state y(∞) = K.

Why Other Options Are Wrong:

• Using s → ∞ is related to initial values, not final values.
• Replacing sY(s) by Y(s) neglects the crucial s multiplier.
• Claims that ignore stability constraints are unsafe and can give incorrect results for oscillatory/unstable systems.

Common Pitfalls:

• Applying FVT to systems with poles on the imaginary axis (e.g., pure integrators or undamped oscillators) leading to wrong or undefined limits.
• Forgetting to include the input when forming Y(s) for closed-loop analysis.

Final Answer:

lim_{t→∞} y(t) = lim_{s→0} s Y(s), with all poles of sY(s) strictly in the left half-plane