Under Countable Choice, the LQR objective is weakly lower semicontinuous

By [A1] and [F4], the weak limit $u$ belongs to ${\mathcal{U}}_c$, so [F2] applies. Put $e_k=x_{u_k}-x_d$ and $e=x_u-x_d$; their difference $e_k-e=x_{u_k}-x_u$ tends to zero uniformly. Since $Q$ is essentially bounded, the estimate $\mathrm{\mid }{a}^{T}Q(t)a-b^{T}Q(t)b\mathrm{\mid }\le \mathrm{\parallel }Q(t){\mathrm{\parallel }}_2(\mathrm{\parallel }a{\mathrm{\parallel }}_2+\mathrm{\parallel }b{\mathrm{\parallel }}_2)\mathrm{\parallel }a-b{\mathrm{\parallel }}_2$ shows that ${\int }_0^T{e}_k^{T}Q{e}_{k}dt\to {\int }_0^T{e}^{T}Qedt$. The terminal terms converge by the same estimate with $P$ and $x_{u_k}(T)\to x_u(T)$.

Let $\mathcal{R}v=R(\cdot )v(\cdot )$. The bounds on $R$ imply $\mathrm{\parallel }\mathcal{R}v{\mathrm{\parallel }}_2\le M\mathrm{\parallel }v{\mathrm{\parallel }}_2$ and $⟨\mathcal{R}v,v⟩\ge 0$; symmetry gives $⟨\mathcal{R}v,w⟩=⟨v,\mathcal{R}w⟩$. For every fixed $w\in L^2$, $⟨\mathcal{R}{u}_k,u_k⟩\ge 2⟨\mathcal{R}{u}_k,w⟩-⟨\mathcal{R}w,w⟩$, because the difference is $⟨\mathcal{R}(u_k-w),u_k-w⟩\ge 0$. The right-hand side converges to $2⟨\mathcal{R}u,w⟩-⟨\mathcal{R}w,w⟩$ by weak convergence, as [F3] makes $v\mapsto ⟨\mathcal{R}v,w⟩=⟨v,\mathcal{R}w⟩$ bounded. Taking the supremum over $w$ gives ${\mathrm{lim\hspace{0.17em}inf}}_k⟨\mathcal{R}{u}_k,u_k⟩\ge ⟨\mathcal{R}u,u⟩$, with equality in the supremum at $w=u$.

The state and terminal cost terms converge by step 1.1, while the control quadratic term is lower semicontinuous by step 1.2. Multiplying by $1\mathrm{/}2$ and adding these three terms gives $J(u)\le {\mathrm{lim\hspace{0.17em}inf}}_{k}J(u_k)$.