Definition:Minimum Value of Functional

Definition
Let $ J \left [ { y } \right ]$ be a functional.

Let $ J$ have a (relative) extremum for $ y= \hat { y }$.

Suppose, $ J \left [ { y } \right ] -  J \left [ \hat { y }  \right ] \ge 0 $ in the neighbourhood of $ y= \hat { y }$.

Then this extremum is called the minimum of the functional $ J$.