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A pointwise L2 box is nonempty, bounded, closed, and convex

Statement

Let $E$ be as in Closed pointwise box of L2 controls, and let $q_{a}, q_{b} \in L^{2} (E; R)$ satisfy $q_{a} \leq q_{b}$ almost everywhere. Then $Q_{a d} (q_{a}, q_{b})$ is nonempty, convex, norm closed, and bounded in $L^{2} (E)$ .

Facts & Assumptions

Given: The finite-measure control domain $E$ , bounds $q_{a}, q_{b} \in L^{2} (E)$ with $q_{a} \leq q_{b}$ almost everywhere, and the definition of $Q_{a d}$ .

[A1]

$Q_{a d} = {q \in L^{2} (E) : q_{a} \leq q \leq q_{b} a.e.}$ (Closed pointwise box of L2 controls).

Proof

1.1A1givenalgebra

The control $q_{a}$ belongs to $Q_{a d}$ because $q_{a} \leq q_{a} \leq q_{b}$ almost everywhere, so $Q_{a d}$ is nonempty. If $q, r \in Q_{a d}$ and $0 \leq t \leq 1$ , then $q_{a} \leq t q + (1 - t) r \leq q_{b}$ almost everywhere; hence the convex combination is in $Q_{a d}$ . The pointwise inequality also gives $∣ q ∣ \leq ∣ q_{a} ∣ + ∣ q_{b} ∣$ almost everywhere for each $q \in Q_{a d}$ , so $∥ q ∥_{L^{2}} \leq ∥ q_{a} ∥_{L^{2}} + ∥ q_{b} ∥_{L^{2}}$ and the box is bounded.

2.1A1givenalgebra∎

Suppose $q_{n} \in Q_{a d}$ and $q_{n} \to q$ in $L^{2} (E)$ . For almost every $x$ , $q_{n} (x) \geq q_{a} (x)$ , so $(q_{a} - q)^{+} \leq ∣ q_{n} - q ∣$ almost everywhere, where $s^{+} = \max {s, 0}$ . Therefore $∥ (q_{a} - q)^{+} ∥_{2} \leq ∥ q_{n} - q ∥_{2} \to 0$ and $q \geq q_{a}$ almost everywhere. Similarly $(q - q_{b})^{+} \leq ∣ q - q_{n} ∣$ gives $q \leq q_{b}$ almost everywhere. Thus $q \in Q_{a d}$ and the box is norm closed.

Depends on

Closed pointwise box of L2 controls

Used by

Existence, uniqueness, and variational inequality for elliptic PDE control Theorem

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

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