Definition:Twice Differentiable/Functional/Dependent on N functions

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Definition

Let $\Delta J \sqbrk {\mathbf y; \mathbf h}$ be an increment of a functional, where $\mathbf y = \paren {\sequence {y_i}_{1 \mathop \le i \mathop \le N} }$ is a vector.

Let:

$\Delta J \sqbrk {\mathbf y; \mathbf h} = \phi_1 \sqbrk {\mathbf y; \mathbf h} + \phi_2 \sqbrk {\mathbf y; \mathbf h} + \epsilon \size {\mathbf h}^2$

where:

$(1): \quad \phi_1 \sqbrk {\mathbf y; \mathbf h}$ is a linear functional

$(2): \quad \phi_2 \sqbrk {\mathbf y; \mathbf h}$ is a quadratic functional with respect to $\mathbf h$

$(3): \quad \ds \size {\mathbf h} = \sum_{i \mathop = 1}^N \size {h_i}_1 = \sum_{i \mathop = 1}^N \max_{a \mathop \le x \mathop \le b} \set {\size {\map {h_i} x} + \size {\map {h_i'} x} }$

$(4): \quad \epsilon \to 0$ as $\size {\mathbf h} \to 0$.

Then the functional $J \sqbrk {\mathbf y}$ is twice differentiable.

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The linear part $\phi_1$ is the first variation, denoted:

$\delta J \sqbrk {\mathbf y; \mathbf h}$

$\phi_2$ is called the second variation (or differential) of a functional, and is denoted:

$\delta^2 J \sqbrk {\mathbf y; \mathbf h}$

Sources

• 1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 5.29$: Generalization to n Unknown Functions