C.3. Liapounov Functions And Global Stability

While the linear stability theory addresses, in principle, the local stability of an infinitesimal region around a stationary solution, the Liapounov criteria deal with global stability of a finite region around a stationary solution.

 

To be more specific, consider a steady state solution y ¯  to eq(1) of section C.1.

We say y ¯  is stable if there exists a function V( y )  such that

1.         V( y ¯ )=0

2.         dV dt 0         yD

3.         V( y )>0     yD

where D={    y   |    | y j y ¯ j |<h      j   }  is a hypercubic region of sides 2h centered at y ¯ .  Conditions (1-3) are called the Liapounov criteria and V a Liapounov fuction .

 

Thus, if the Liapounov criteria are satisfied, any trajectory that falls inside D once will forever remain in it.  To see how this comes about, consider 2 more hypercubes centered at y ¯  with sides 2e and 2d such that δ<ε<h .  Let B δ , B ε , B h  be the boundaries of these hypercubes.  Denoting the lower bound of V( y )  on B ε  by a, we have Vα>0  on B ε .  Now, consider a point y 0 =y( t 0 )  inside the boundary B δ  at t= t 0 .  Let y t =y( t )  be the position of the point at time t.  If B δ  is chosen such that V( y 0 )<α , then, according to criterion (2), we have

        V( y t )<V( y 0 )<α

so that the point must remain inside B ε  for all t.

 

An analogous argument can be made for the case V( y )<0  and dV dt 0  so that y ¯  is again stable if such a function can be found.

Exercise C.2

Consider a damped oscillator described by

        d 2 y 1 d t 2 +α d y 1 dt + ω 0 2 y 1 =0                   (1a)

(a)     Show that the total energy E= 1 2 ( y ˙ 1 2 + ω 0 2 y 1 2 )  can be used as a Lyapounov function.

(b)     Locate and classify the steady states for the cases α>0  and α=0 .

Answer (a)

The equivalent set of 1st order equations can be found by introducing a new variable y 2 = y ˙ 1 +α y 1  so that

        y ˙ 1 = y 2 α y 1                                    (1b)

and, with the help of (1a),

y ˙ 2 = y ¨ 1 +α y ˙ 1 = ω 0 2 y 1                      (1c)

Consider now the real function

        F( y 1 , y 2 )= 1 2 ( y 2 2 + ω 0 2 y 1 2 )>0          (2)

we have

        dF dt = y 2 y ˙ 2 + ω 0 2 y 1 y ˙ 1   = y 2 ω 0 2 y 1 + ω 0 2 y 1 ( y 2 α y 1 )   =α ω 0 2 y 1 2         (3)

Thus, for α0 , we have dF dt 0  so that F is a Lyapounov function.  In particular, for α=0 , the system is non-dissipative and F=E .

Answer (b)

A steady state to (1b,c) is y 1 = y 2 =0 .  For α0 , F is a Lyapounov function for all the space around y=( y 1 , y 2 )=0 .  Hence, y=0  is a stable.  To be more precise, for α>0 , it is a stable focus.  For α=0 , it is a center.