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The optimal elliptic control is the clipped negative adjoint

Statement

For the unique minimizer $q_{*}$ in Existence, uniqueness, and variational inequality for elliptic PDE control, with adjoint $p (q_{*})$ , one has

q_{*} (x) = \min {q_{b} (x), \max {q_{a} (x), - \frac{B^{*} p (q_{*}) (x)}{α}}} for almost every x \in E .

Facts & Assumptions

Given: The linear-quadratic reduced objective, the nonempty closed convex box $Q_{a d}$ , its unique minimizer $q_{*}$ , and $α > 0$ .

[F1]

The minimizer is characterized by $(B^{*} p (q_{*}) + α q_{*}, v - q_{*})_{U} \geq 0$ for every $v \in Q_{a d}$ (Existence, uniqueness, and variational inequality for elliptic PDE control).

[F2]

The unique $L^{2}$ projection onto $Q_{a d}$ is pointwise clipping and is characterized by its variational inequality (The L2 projection onto a pointwise box is clipping).

Proof

1.1F1F2givenalgebra

Divide the variational inequality in [F1] by $α > 0$ and rewrite it as ${(q_{*} - (- B^{*} p (q_{*}) / α), v - q_{*})}_{U} \geq 0 (v \in Q_{a d}) .$ This is exactly the projection variational inequality in [F2] for projecting $- B^{*} p (q_{*}) / α$ onto $Q_{a d}$ .

2.1F2step 1.1∎

By the explicit formula in [F2], the projection is pointwise clipping to $[q_{a} (x), q_{b} (x)]$ almost everywhere. Substitution gives the displayed formula.

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Sources

Eduardo Casas and Fredi Tröltzsch, Error estimates for linear-quadratic elliptic control problems (standard reference, not scraped)