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The method of investigating the stability using Liapunov function is _____________

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Answer & Explanation

Correct AnswerA. The direct method

Quick Explanation

Investigating stability by constructing a Lyapunov function V(x) and examining its derivative along system trajectories is called Lyapunov's direct (second) method. It avoids explicitly solving the nonlinear differential equations.

Formula / Key Relation

V(x)>0 for x≠0, V(0)=0

V̇(x)<0 ⇒ asymptotic stability (standard local/global conditions apply)

Detailed Explanation

Concept & Reasoning

A suitable Lyapunov function is positive definite near the equilibrium and its time derivative is negative definite for asymptotic stability (with standard qualifications). The indirect method instead linearizes the system near an equilibrium and inspects the linearized eigenvalues.

How to Solve It in the Exam

Lyapunov function itself → direct method.

Important Exam Point

Indirect method = linearization/eigenvalues.

Related Revision Path

This question sits in the revision path Control Systems → Nonlinear Control → Lyapunov Stability .

Direct Lyapunov Method explanatory diagram

Quick Trick

Lyapunov function itself → direct method.

Common Mistake

Calling it indirect merely because V̇ is computed indirectly from dynamics.

Exam Tip

Indirect method = linearization/eigenvalues.

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