Linear coercive elliptic PDE control problem and reduced objective

Let $\mathrm{\Omega }\subset {\mathbb{R}}^d$ be a bounded Lipschitz domain, $V=H_0^1(\mathrm{\Omega };\mathbb{R})$, and $H=L^2(\mathrm{\Omega };\mathbb{R})$. Use the real integral pairing on $H$. Equip $V$ with the standard full Sobolev inner product and norm $(v,w{)}_V=(v,w{)}_H+{\sum }_{i=1}^d(D_{i}v,D_{i}w{)}_H,\mathrm{\parallel }v{\mathrm{\parallel }}_V^2=\mathrm{\parallel }v{\mathrm{\parallel }}_H^2+{\sum }_{i=1}^d\mathrm{\parallel }{D}_{i}v{\mathrm{\parallel }}_H^2\mathrm{.}$ Here $H_0^1=W_0^{1,2}$ is the closure of $C_c^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{R})$ in the $W^{1,2}$ norm of Zero-boundary Sobolev space as a norm closure. Under Countable Choice, $H$ and $V$ are real Hilbert spaces; the local proof of completeness of $V$ is included in Well-posedness of elliptic control state and adjoint equations ↗. The inclusion $J:V\to H$ is continuous with $\mathrm{\parallel }J\mathrm{\parallel }\le 1$.

Let $E\subseteq \mathrm{\Omega }$ be bounded and measurable, $U=L^2(E;\mathbb{R})$ with its real integral inner product, and let $Q_{ad}\subset U$ be the pointwise box of Closed pointwise box of L2 controls. Under Countable Choice, $U$ is a real Hilbert space by $L^2$ with the integral pairing is a Hilbert space. Let $a:V\times V\to \mathbb{R}$ be bilinear, bounded with constant $M$, and coercive with constant $\beta >0$ in the stated $V$ norm; symmetry is not assumed. Let $B:U\to V^{\ast }$ be bounded and linear, and fix $f\in V^{\ast }$, $u_d\in H$, and $\alpha >0$.

For every $q\in U$, the state $u(q)\in V$ is required to satisfy

$$a(u(q),v)=⟨f+Bq,v{⟩}_{V^{\ast },V}(v\in V)\mathrm{.}$$

The reduced objective, defined on all of $U$ and minimized over $Q_{ad}$, is

$$\widehat{J}(q)=\frac{1}{2}\mathrm{\parallel }Ju(q)-u_d{\mathrm{\parallel }}_H^2+\frac{\alpha }{2}\mathrm{\parallel }q{\mathrm{\parallel }}_U^2\mathrm{.}$$

A spatially varying coefficient, boundary control, variational crimes, finite-element control discretization, and parabolic evolution are separate hypotheses or branches; none is implicit in this definition. The state well-definedness is supplied by Well-posedness of elliptic control state and adjoint equations ↗.