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Existence, uniqueness, and variational inequality for elliptic PDE control

Statement

Assume Countable Choice and the setting of Linear coercive elliptic PDE control problem and reduced objective. Then $\hat{J}$ has a unique minimizer $q_{*} \in Q_{a d}$ . It is characterized by

(B^{*} p (q_{*}) + α q_{*}, v - q_{*})_{U} \geq 0 (v \in Q_{a d}),

where $p (q_{*})$ is the adjoint from Well-posedness of elliptic control state and adjoint equations. Conversely, every $q \in Q_{a d}$ satisfying this variational inequality is the unique minimizer.

Facts & Assumptions

Given: Countable Choice, the nonempty pointwise box $Q_{a d} \subset U$ , the bounded coercive state problem, and the reduced objective $\hat{J}$ .

[F1]

$Q_{a d}$ is nonempty, bounded, norm closed, and weakly sequentially closed (A pointwise L2 box is nonempty, bounded, closed, and convex, A pointwise L2 box is weakly sequentially closed).

[F2]

Every bounded sequence in the separable real Hilbert control space $U = L^{2} (E)$ has a weakly convergent subsequence (A bounded sequence in a separable Hilbert space has a weakly convergent subsequence, L2 controls on a bounded spatial region form a separable Hilbert space).

[F3]

The state map is affine with bounded linear part $S$ , and $\hat{J} (q + h) - \hat{J} (q) = (B^{*} p (q) + α q, h)_{U} + \frac{1}{2} ∥ S h ∥_{H}^{2} + \frac{α}{2} ∥ h ∥_{U}^{2}$ (Reduced gradient and coercive Hessian for linear elliptic tracking control).

[F4]

A Fréchet minimizer over a convex set has nonnegative directional derivative along every feasible segment; here that derivative is $(B^{*} p (q) + α q, v - q)_{U}$ (Reduced gradient and coercive Hessian for linear elliptic tracking control).

[A1]

Countable Choice permits the indexed selection of a minimizing sequence from the nonempty sets ${q \in Q_{a d} : \hat{J} (q) < m + 1 / (n + 1)}$ for $n \in N$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

1.1A1F1F2givenconstruct

The box is nonempty by [F1], so $m : = \inf_{q \in Q_{a d}} \hat{J} (q)$ is finite: $\hat{J} \geq 0$ and $\hat{J} (q_{a}) < \infty$ . For every $n \in N$ , the defining property of the infimum makes the set in [A1] nonempty. By Countable Choice choose $q_{n} \in Q_{a d}$ with $\hat{J} (q_{n}) < m + 1 / (n + 1)$ . Since $\hat{J} (q_{n}) \geq α ∥ q_{n} ∥_{U}^{2} / 2$ and $1 / (n + 1) \leq 1$ , the sequence is bounded. By [F2], applied to its positive-indexed tail, a subsequence converges weakly to some $q \in U$ , and [F1] gives $q \in Q_{a d}$ .

1.2F4givenalgebra

If $q_{*}$ minimizes, fix $v \in Q_{a d}$ . Convexity gives $q_{*} + t (v - q_{*}) \in Q_{a d}$ for $0 \leq t \leq 1$ . Minimality makes the right directional quotient nonnegative for every $t > 0$ ; letting $t ↓ 0$ and using [F4] proves the variational inequality.

2.1F1F3step 1.1algebra

Write $J u (q_{n}) - u_{d} = J u (0) - u_{d} + S q_{n}$ . Bounded linear maps preserve weak convergence, so along the subsequence this state residual converges weakly in $H$ to $J u (q) - u_{d}$ ; the controls converge weakly in $U$ . The Hilbert norm is weakly lower semicontinuous: if $z \neq 0$ , then $∥ z_{n} ∥ \geq ∣ (z_{n}, z / ∥ z ∥) ∣ \to ∥ z ∥$ , while the case $z = 0$ is immediate. Applying this to both squared norms in $\hat{J}$ gives $\hat{J} (q) \leq {lim inf}_{n} \hat{J} (q_{n}) = m$ . Thus $q$ attains the infimum.

3.1F1F3step 2.1algebra

The exact expansion in [F3] shows strict convexity: for distinct $q, r \in U$ and $0 < t < 1$ , $\hat{J} (t q + (1 - t) r) = t \hat{J} (q) + (1 - t) \hat{J} (r) - \frac{t (1 - t)}{2} (∥ S (q - r) ∥_{H}^{2} + α ∥ q - r ∥_{U}^{2}) .$ The last subtracted term is positive, so two different minimizers in the convex box cannot exist.

4.1F3step 3.1step 1.2givenalgebra∎

Conversely, if $q$ satisfies the variational inequality, then for every $v \in Q_{a d}$ the exact expansion [F3] gives $\hat{J} (v) - \hat{J} (q) = (B^{*} p (q) + α q, v - q)_{U} + \frac{1}{2} ∥ S (v - q) ∥_{H}^{2} + \frac{α}{2} ∥ v - q ∥_{U}^{2} \geq 0.$ Thus $q$ is a minimizer; step 3.1 gives uniqueness.

Remarks

The existence proof uses Countable Choice to select a minimizing sequence and the stated separable-Hilbert weak subsequence lemma. It uses no Axiom of Choice, Hahn–Banach separation, or reflexive-space compactness theorem.

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Sources

Hannes Meinlschmidt, Optimization with Partial Differential Equations, FAU lecture notes (standard reference, not scraped)