Contribution to Functional Change

The functional change, I yo + h) — I yo), can be expressed in terms of the second variation provided (d /da )/(yo+a/i) exists and is continuous at a = 0 for all h. Using the second order Taylor expansion (Appendix 2.A, p. 52) along 

Since 5I yo g) and S I yo, g) are homogeneous of the respective first and second degrees, 

The above result can be generalized to a functional dependent on several functions. Thus, 

We end this chapter by pointing out that the variation of an objective functional will provide us with important clues about its optimum, similar to what a differential does for an objective function. The second variation will provide some auxiliary conditions and help in the search for optimal solutions. The necessary and sufficient conditions for the optimum of an objective functional will be the topic of the next chapter. As expected, those conditions will use the concepts we have developed here. 

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