Weak Lower Semicontinuity of the Running-Cost Integral along a State Path

Let $l$ and $m$ be natural numbers with $l\ge2$ and $m\ge1$, let $(L,G)$ be population cost data on $l$ states with control dimension $m$, and let $\Delta^{l}$ be the probability simplex. Let $T>0$ be a real number, adopt the notation of the Lebesgue space $L^{2}([0,T];\mathbb{R}^{d})$ in the case $d=m$, write $H=L^{2}([0,T];\mathbb{R}^{m})$ with its pairing, norm $\lVert\cdot\rVert_{L^{2}}$ and metric $d_{L^{2}}$, and write $\lambda=\lambda_{[0,T]}$ and $\mathcal{B}=\mathcal{B}_{[0,T]}$. Write $|\cdot|$ for the Euclidean norm on $\mathbb{R}^{k}$, for any natural number $k\ge1$, and for the absolute value on $\mathbb{R}$.

Let $\mathcal{A}$ be a nonempty convex subset of $\mathbb{R}^{m}$ that is compact for the topology determined by the Euclidean distance, and let $\mathcal{U}_{\mathcal{A}}$ be the set of $\mathcal{A}$-valued controls. Assume in addition that $L$ is convex in the control on $\mathcal{A}$, that is, that for every $\Sigma\in\Delta^{l}$ the function $\mathcal{A}\to\mathbb{R}$ sending $a$ to $L(\Sigma,a)$ is convex on $\mathcal{A}$.

Call a map $S:[0,T]\to\mathbb{R}^{l}$ an admissible state path if $S_{t}\in\Delta^{l}$ for every $t\in[0,T]$ and each component of $S$ is measurable from $([0,T],\mathcal{B})$ to $\mathbb{R}$ with its Borel $\sigma$-algebra. Then the following hold.

1. (The running-cost integral is well defined.) Let $S$ be an admissible state path and let $\xi\in\mathcal{U}_{\mathcal{A}}$. Then $\xi$ has at least one representative whose values all lie in $\mathcal{A}$; denoting such a representative again by $\xi$, as in claim 5 of the Lebesgue space, the map $t\mapsto L(S_{t},\xi(t))$ is measurable and bounded, hence integrable, and the value

$$\Phi_{S}(\xi)=\int_{[0,T]}L\bigl(S_{t},\xi(t)\bigr)\,d\lambda(t),$$

called the running-cost integral of $\xi$ along $S$, does not depend on the choice of such a representative. Moreover there is a real number $C\ge0$, depending only on $L$, $\Delta^{l}$ and $\mathcal{A}$, with $|\Phi_{S}(\xi)|\le CT$ for every admissible state path $S$ and every $\xi\in\mathcal{U}_{\mathcal{A}}$; consequently, for any sequence $(S^{n})_{n\in\mathbb{N}}$ of admissible state paths and any sequence $(\xi_{n})_{n\in\mathbb{N}}$ in $\mathcal{U}_{\mathcal{A}}$, the real sequence $\bigl(\Phi_{S^{n}}(\xi_{n})\bigr)_{n\in\mathbb{N}}$ is a bounded sequence, so its limit inferior is defined.

2. (Convex closed sublevel sets.) Let $S$ be an admissible state path and let $M$ be a real number. The set $C_{M}=\{\xi\in\mathcal{U}_{\mathcal{A}}:\Phi_{S}(\xi)\le M\}$ is a convex subset of $H$ and is closed for the topology of metric open subsets determined by $d_{L^{2}}$.

3. (Weak lower semicontinuity.) Let $S$ be an admissible state path, let $(\xi_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathcal{U}_{\mathcal{A}}$, let $\xi\in H$ with $\xi_{n}\rightharpoonup\xi$ in the sense of weak convergence. Then $\xi\in\mathcal{U}_{\mathcal{A}}$ and

$$\Phi_{S}(\xi)\le\liminf_{n}\Phi_{S}(\xi_{n}).$$

4. (Uniformly convergent states.) Let $S$ and $S^{n}$ ($n\in\mathbb{N}$) be admissible state paths such that for every real $\varepsilon>0$ there is $N\in\mathbb{N}$ with $|S^{n}_{t}-S_{t}|\le\varepsilon$ for every $t\in[0,T]$ and every $n\ge N$, let $(\xi_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathcal{U}_{\mathcal{A}}$ with $\xi_{n}\rightharpoonup\xi$ for some $\xi\in H$. Then $\xi\in\mathcal{U}_{\mathcal{A}}$ and

$$\Phi_{S}(\xi)\le\liminf_{n}\Phi_{S^{n}}(\xi_{n}).$$