A parabolic adjoint gives the gradient when the state map is differentiable

Assume Countable Choice and $T>0$, and use the real evolution triple $(V,H,V^{\ast })$ of Parameter-dependent parabolic variational states and objectives. On an open parameter set $\mathrm{\Theta }\subseteq {\mathbb{R}}^m$, assume the state map $\theta \mapsto u_{\theta }$ is Fr'echet differentiable into $W(0,T;V)$, with derivative $w_{\eta }$ in direction $\eta \in {\mathbb{R}}^m$. For each fixed $\theta$ and direction $\eta$, assume there is a null set $N_{\theta ,\eta }\subset (0,T)$ such that the linearized equation below holds for every $t\mathrm{\notin }{N}_{\theta ,\eta }$ and simultaneously every $v\in V$, using representatives of its data: $⟨w_{\eta }^{\mathrm{\prime }}(t),v⟩+a_{\theta }(t;w_{\eta }(t),v)=⟨f_{\theta }^{\mathrm{\prime }}[\eta ](t),v⟩-a_{\theta }^{\mathrm{\prime }}[\eta ](t;u_{\theta }(t),v),w_{\eta }(0)=u_{0,\theta }^{\mathrm{\prime }}[\eta ]\mathrm{.}$ Let $\varphi :H\times \mathrm{\Theta }\to \mathbb{R}$ be $C^1$, and let $L:[0,T]\times V\times \mathrm{\Theta }\to \mathbb{R}$ be measurable in time and $C^1$ in $(u,\theta )$ for almost every time, with a common integrable majorant that justifies differentiating its time integral. Assume $D_u\varphi$ is represented by an element of $H$ and $D_{u}L(t,u_{\theta },\theta )\in V^{\ast }$ is square-integrable in time. Suppose the adjoint $p\in W(0,T;V)$ satisfies the following equation simultaneously for every $v\in V$ outside one null set $M_{\theta }\subset (0,T)$, and has the stated terminal trace: $-⟨p^{\mathrm{\prime }}(t),v⟩+a_{\theta }(t;v,p(t))=D_{u}L(t,u_{\theta }(t),\theta )[v],p(T)={\mathrm{\nabla }}_H\varphi (u_{\theta }(T),\theta )\mathrm{.}$ Then for $j(\theta )=\varphi (u_{\theta }(T),\theta )+{\int }_0^{T}L(t,u_{\theta }(t),\theta )dt$, $Dj(\theta )[\eta ]=D_{\theta }\varphi (u_{\theta }(T),\theta )[\eta ]+(p(0),u_{0,\theta }^{\mathrm{\prime }}[\eta ]{)}_H+{\int }_0^T(D_{\theta }L(t,u_{\theta },\theta )[\eta ]+⟨f_{\theta }^{\mathrm{\prime }}[\eta ](t),p(t)⟩-a_{\theta }^{\mathrm{\prime }}[\eta ](t;u_{\theta }(t),p(t)))dt\mathrm{.}$ The exceptional sets are independent of the test $v$; they need not be common to all parameters or directions. The forcing/form-derivative difference in the integrand is interpreted together. The hypotheses imply it is measurable and integrable: almost everywhere it equals $⟨w_{\eta }^{\mathrm{\prime }},p⟩+⟨p^{\mathrm{\prime }},w_{\eta }⟩+D_{u}L[w_{\eta }]$. No global boundedness or coercivity of $a_{\theta }$, or separability of $V$, is required for this conditional identity.

This is conditional on the displayed differentiable state and adjoint; it does not prove either exists. The adjoint form is transposed as shown, so symmetry of $a_{\theta }$ is not required.