Parameter-dependent parabolic variational states and objectives

Let $T>0$, let $(V,H,V^{\ast })$ be the real evolution triple and let $\mathrm{\Theta }\subseteq {\mathbb{R}}^m$ be open. A parameter-dependent parabolic variational problem specifies, for each $\theta \in \mathrm{\Theta }$, a bilinear form $a_{\theta }(t;\cdot ,\cdot )$ and data $f_{\theta }\in L^2(0,T;V^{\ast })$, $u_{0,\theta }\in H$. A state is a function $u_{\theta }\in W(0,T;V)$ with continuous $H$ trace, $u_{\theta }(0)=u_{0,\theta }$, and $⟨u_{\theta }^{\mathrm{\prime }}(t),v{⟩}_{V^{\ast },V}+a_{\theta }(t;u_{\theta }(t),v)=⟨f_{\theta }(t),v{⟩}_{V^{\ast },V}$ for every $v\in V$ and almost every $t$. When a state is specified, let $\varphi :H\times \mathrm{\Theta }\to \mathbb{R}$ be differentiable and let $L:(0,T)\times V\times \mathrm{\Theta }\to \mathbb{R}$ be differentiable in $(v,\theta )$ for almost every $t$. Require $t\mapsto L(t,u_{\theta }(t),\theta )$ to be measurable and to belong to $L^1(0,T)$ for every parameter under consideration. Then the finite real number $j(\theta )=\varphi (u_{\theta }(T),\theta )+{\int }_0^{T}L(t,u_{\theta }(t),\theta )dt$ is its reduced objective. This definition does not assert existence of a state, parameter differentiability of the state or reduced objective, or optimality.