Necessary Condition for Twice Differentiable Functional to have Minimum

Theorem
Let $J\sqbrk y$ be a twice differentiable functional.

Then $J$ has a minimum for $y=\hat y$ if


 * $\delta^2 J\sqbrk{y;h}\ge 0$

for $y=\hat y$ and all admissible $h$.

Proof
By definition, $\Delta J\sqbrk y$ can be expressed as:


 * $\Delta J\sqbrk{y;h}=\delta J\sqbrk{y;h}+\delta^2 J\sqbrk{y;h}+\epsilon\size {h}^2$

By Condition for Differentiable Functional to have Extremum:


 * $\delta J\sqbrk{\hat y;h}=0$

Hence:


 * $\Delta J \sqbrk{\hat y;h}=\delta^2 J\sqbrk{\hat y;h}+\epsilon\size {h}^2 $

and $\Delta J\sqbrk{\hat y;h}$ and $\delta^2 J\sqbrk{\hat y;h}$ will have the same sign for sufficiently small $\size h$.

Suppose, there exists $h=h_0$ such that:


 * $\delta^2 J\sqbrk{\hat y;h_0}<0$

Then, for any $\alpha\ne 0$

Therefore, $\Delta J\sqbrk{\hat y;h}$ can be made negative for arbitrary small $\size h$.

However, by assumption $\Delta J\sqbrk{\hat y;h}$ is a minimum of $\Delta J\sqbrk{y;h}$ for all sufficiently small $\size h$.

This is a contradiction.

Thus, a function $h_0:\delta^2 J\sqbrk{\hat y;h_0}<0$ does not exist.

In other words:


 * $\delta^2 J\sqbrk{\hat y;h}\ge 0$

for all $h$.