Liapunov’s Stability Critearia

The system is stable throughout the region where dV/dt is negative-semidefinite for all nonzero x and t>0
The system is asymtotically stable in the region if dV/dt is negative-definite
The system is globally asymtotically stable if dV/dt is negative-definite in the entire state space governed by the variable x

Example :

1.

x^{'}_{1}=- x_{2}- x^{3}_{1}

x^{'}_{2}=+ x_{1}- x^{3}_{2}

V\left({x}_{1},{x}_{2} \right)= x^{2}_{1}- x^{2}_{2}

W\left(x \right)=dV/dx=\left(\delta V/\delta {x}_{1} \right)d{x}_{1}+\left(\delta V/\delta {x}_{2} \right)d{x}_{2}

W=2{x}_{1}{x}^{'}_{1}+2{x}_{2}{x}^{'}_{2}=-2\left({x}^{4}_{1}+{x}^{4}_{2} \right)

Since V > 0,       V(0)=0,    V is positive-definite

Since W < 0,       V(0)=0,    V is negative-definite

2. Linear Oscillator

{y}^{\"}+y=0

{x}^{'}_{1}={x}_{2}

{x}^{'}_{2}=-{x}_{1}

V\left({x}_{1},{x}_{2} \right)=\left({x}^{2}_{1}+{x}^{2}_{2} \right)

W\left({x}_{1},{x}_{2} \right)=0

Hence, the system is globally stable

2.
{x}^{'}_{1}={x}_{2}-a{x}_{1}\left({x}^{2}_{1}+{x}^{2}_{2} \right)

{x}^{'}_{2}={x}_{1}-a{x}_{2}\left({x}^{2}_{1}+{x}^{2}_{2} \right)

V\left({x}_{1},{x}_{2} \right)=\left({x}^{2}_{1}+{x}^{2}_{2} \right)

W\left({x}_{1},{x}_{2} \right)=-2a{V}^{2}

Since V(0)=0,         V>0,           W<0,         \lim_{x\rightarrow \sim }V=\sim

The system is globally asymtotically stable.

Source:

Kolk, W. Richard, Lerman, Robert A., Nonlinear System Dynamics, van Nostrand Reinhold, USA, page 191,1992.

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