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Which one of the following is not a property of the Liapunov function?
- a)It is a unique function for a given system.
- b)It is positive definite, atleast in the neighbourhood of the origin.
- c)It is a scalar function.
- d)It’s time derivative is non-positive.
Correct answer is option 'A'. Can you explain this answer?
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Which one of the following is not a property of the Liapunov function?...
To determine which of the given properties is not true for a Liapunov function, let's analyze each option:
A: It is a unique function for a given system. This statement is true. A Liapunov function is unique for a given system. If there were multiple Liapunov functions for the same system, it would imply different stability properties, which would contradict the definition of a Liapunov function.
B: It is positive definite, at least in the neighborhood of the origin. This statement is generally true. A Liapunov function must be positive definite in a neighborhood of the origin, meaning it is greater than zero for all points other than the origin within a certain distance. This condition ensures that the function is decreasing towards the equilibrium point.
C: It is a scalar function. This statement is true. A Liapunov function is a scalar function, meaning it takes a single value for each point in the system's state space.
D: Its time derivative is non-positive. This statement is also true. The time derivative of a Liapunov function must be non-positive for all points within the system's state space, except at the origin. This condition guarantees that the function is decreasing along the system's trajectories towards the equilibrium point.
Therefore, the property that is not true for a Liapunov function is option A: It is not a unique function for a given system.
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Which one of the following is not a property of the Liapunov function?a)It is a unique function for a given system.b)It is positive definite, atleast in the neighbourhood of the origin.c)It is a scalar function.d)It’s time derivative is non-positive.Correct answer is option 'A'. Can you explain this answer?
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