Question

Given below are two statements:
Assertion (A): Lyapunov direct method states that the equilibrium point of a system is asymptotically stable if there exists a positive definite function whose derivative is negative definite.
Reason (R): The equilibrium point of \(\dfrac{dx}{dt}=-y,\ \dfrac{dy}{dt}=x\) is asymptotically stable.

• A. Both (A) and (R) are correct and (R) is the correct explanation of (A)
• B. Both (A) and (R) are correct but (R) is not the correct explanation of (A)
• C. (A) is correct but (R) is not correct
• D. (A) is not correct but (R) is correct

Hint:

For asymptotic stability, \(\dot{V}\) must be negative definite, not merely zero.

Answer 1

The Correct Option is C

Concept:
Lyapunov direct method is used to test stability of equilibrium points.

Step 1: Check Assertion.
If a positive definite Lyapunov function \(V\) exists and its derivative \(\dot{V}\) is negative definite, then the equilibrium point is asymptotically stable. So: \[ A \text{ is correct} \]

Step 2: Check Reason.
For: \[ \frac{dx}{dt}=-y,\qquad \frac{dy}{dt}=x \] Choose: \[ V=x^2+y^2 \] Then: \[ \dot{V}=2x\frac{dx}{dt}+2y\frac{dy}{dt} \] \[ \dot{V}=2x(-y)+2y(x)=0 \] Since \(\dot{V}=0\), the motion is stable but not asymptotically stable. Thus: \[ R \text{ is incorrect} \] \[ \therefore \text{Correct Answer is (C)} \]