Bound on Possible Transient Lengths

A slightly different Lyapunov function from the one defined above can be used to put bounds on the transient length. Let 

A simple calculation shows that AjL = L [S(t),5 t + 1)] — L [S(t — l),o(t)] = 4AtL, so that if L is a decreasing Lyapunov function, L must also be a decreasing Lyapunov function.  

We introduce two useful quantities (1) From the definition of L (equation 5.123), we know that L is strictly decreasing for all o G Xr. We can therefore define the positive quantity 

To see how to use this result to bound the transient length, we first note that since L is strictly decreasing in the transient phase, equation 5.129 yields 

Therefore, using equation 5.135 and setting t = t, we have that 

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