Rules of the Calculus of Variations

The manipulations used to derive the Adjoint State Method.

Definitions.

• A functional $J[u]$ maps a function $u$ to a scalar.
• An admissible variation $\delta u$ keeps $u+\epsilon \delta u$ admissible for small $\epsilon$. It satisfies the homogeneous version of all essential boundary conditions, for example $\delta u=0$ where $u$ is prescribed.
• The first variation (Gâteaux derivative) is $J^{\prime }[u](\delta u)=\frac{\mathrm{d}}{\mathrm{d}\epsilon }J[u+\epsilon \delta u]{|}_{\epsilon =0}.$

Rules.

• Linearity: $\delta (aF+bG)=a\delta F+b\delta G$.
• Product and chain rule: $\delta (FG)=(\delta F)G+F(\delta G)$, $\delta \Phi (u)={\Phi }^{\prime }(u)\delta u$. Only terms of first order in $\epsilon$ are kept.
• Commutation: on a fixed domain, variation commutes with derivatives and integrals: $\delta \nabla u=\nabla \delta u$ and $\delta {\int }_{\Omega }F={\int }_{\Omega }\delta F$. On a moving domain, a Leibniz or Reynolds transport term appears (shape derivatives).
• Fundamental lemma: if ${\int }_{\Omega }F\eta =0$ for all $\eta \in C_c^{\infty }(\Omega )$, then $F=0$ a.e. Derivatives must first be moved off $\eta$ by integration by parts.
• Boundary terms decide boundary conditions: at an essential condition, $\delta u=0$, so the term drops and gives no condition on $\lambda$. At a natural (free) boundary, $\delta u$ is arbitrary, so its coefficient must vanish, which gives a boundary condition on $\lambda$.
• Independence of variations: for $\mathcal{L}(u,\lambda )$, a joint variation gives $\delta \mathcal{L}={\partial }_u\mathcal{L}\delta u+{\partial }_{\lambda }\mathcal{L}\delta \lambda +\mathcal{O}({\epsilon }^2)$. So ${\partial }_u\mathcal{L}=0$ and ${\partial }_{\lambda }\mathcal{L}=0$ are separate conditions.

Pitfalls. Second variations are only needed to classify stationary points, not for gradients. $\lambda$ is a free test function only until the adjoint equation fixes it.

References

1. I. M. Gelfand, S. V. Fomin (1963). Calculus of Variations. Prentice-Hall (Dover reprint 2000). ISBN 978-0-486-41448-5
2. M. Hinze, R. Pinnau, M. Ulbrich, S. Ulbrich (2009). Optimization with PDE Constraints. Springer. doi:10.1007/978-1-4020-8839-1