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In this method, we find the Lyapunov function in backward direction by assuming  and then calculate  [Normally we assumed V(X) and then evaluated]

Consider a nonlinear, autonomous and time invariant system   

                                                                   (1)

         ; X

The standard procedure to find V(X)

1.  Let  be a scalar function of X, and the vector function g(X)

_nx1 (2)

If V(X) is selected, then   , and g(X) to be calculated

2. (3)

To get integrate (3)

              (4)

Note 1: However, the integral value depends on the initial and final states (not on the path followed). Hence, integration can be convenient done along each of the co-ordinate axes in turn.

                    (5)

Note 2: The free parameter of  are constrained to the symmetric condition, which is satisfied by all gradients of a scalar functions.

Similarly,

Example 1: Analyze the stability behavior of the following system

a>0, b<0

Solution:  Let’s assume X  is an equilibrium point,

Assume

Note: , Assume ‘0’ without loss of  symmetry

Further,

Now,

Choose , Then  (PDF)

 is a Lyapunov function candidate.

Now        

Let us choose

Unless we know about a, b at this point nothing can be said about

• Let us assume . Then,

 in some domain  and

i.e.,  is negative definite in D.

The system is locally asymptotically stable.

Example 2: Analyze the stability behavior of the following system by using variable gradient method (same as previous example, now assume a=1,b=-1)

Solution:  Let’s assume X  is an equilibrium point,

Assume

Note: , Assume ‘0’ without loss of  symmetry

Further, =g(X)

Now,

Choose , Then  (PDF)

 is a Lyapunov function candidate.

Let us choose

Thus, if , then  is negative definite.

 in some domain 1>  i.e.  and

i.e.,  is negative definite in D.

The system is locally asymptotically stable.

• V(X) is a positive definite function: Lyapunov function and the systems is locally asymptotic stability of the origin in the region  is guaranteed.