Convergence of Corrections and Costs along Approximating Sequences of an Extended Admissible Control

Throughout, a real-valued function on a subinterval $I$ of the real numbers $\mathbb{R}$ is called continuous on $I$ when it is continuous relative to $I$, both $I$ and the codomain $\mathbb{R}$ carrying the metric of the real line.

Consider a linear-Gaussian state-observation model on $[0,T]$, a control dimension $k\ge1$, and a control matrix assignment $B$ as in Controlled State and Controlled Observations in the Linear-Gaussian Model, with notation and fixed versions as in those items; adopt the distance $d$ of Almost-Everywhere Mean-Square Limits of Cauchy Sequences of Admissible Controls and the mean-square norm $\lVert\cdot\rVert_{2}$ of Square-Integrable Random Variables and the Mean-Square Inner Product. Then:

1. (Consistency of the class) Every admissible control $\beta$ with values in $\mathbb{R}^{k}$ is an extended admissible control with values in $\mathbb{R}^{k}$, and the constant sequence $\beta^{(n)}:=\beta$ together with $D:=[0,T]$ is an approximating sequence for it.

Now let $\alpha$ be an extended admissible control with values in $\mathbb{R}^{k}$ and let $\bigl((\alpha^{(n)}),D\bigr)$ be an approximating sequence for $\alpha$. For each $n$ let $c^{(n)}$ denote the correction process of Superposition Decomposition of the Controlled State and Observations for the admissible control $\alpha^{(n)}$, with fixed versions as there.

2. (Convergence of the corrections) There exists a family $c^{\alpha}=(c^{\alpha}_t)_{t\in[0,T]}$ of tuples $c^{\alpha}_t=(c^{\alpha,1}_t,\dots,c^{\alpha,l}_t)$ of square-integrable random variables such that:

(a) each component family $(c^{\alpha,i}_t)_{t\in[0,T]}$ is mean-square continuous, $c^{\alpha,i}_0=0$ almost surely, and for every $t\in[0,T]$ each $c^{\alpha,i}_t$ is almost surely equal to a $\mathcal{G}_t$-measurable square-integrable random variable, $\mathcal{G}_t$ being the observation $\sigma$-algebras of the model;

(b) the function $t\mapsto\sum_{i=1}^{l}\lVert c^{(n),i}_t-c^{\alpha,i}_t\rVert_{2}$ is continuous on $[0,T]$ for each $n$, its maximum over $[0,T]$ exists by Extreme Value Theorem on a Closed Real Interval, and these maxima tend to $0$ as $n\to\infty$;

(c) if $\bigl((\beta^{(n)}),D'\bigr)$ is any approximating sequence for $\alpha$ (possibly the same one), with correction processes $b^{(n)}$, and $\tilde c$ is any family satisfying (a) and (b) for that sequence, then for every $t\in[0,T]$ and every $i$: $\tilde c^{\,i}_t=c^{\alpha,i}_t$ almost surely; in particular, if $\alpha$ is itself admissible, then $c^{\alpha,i}_t$ is almost surely equal to the $i$-th component at $t$ of the correction process of Superposition Decomposition of the Controlled State and Observations for $\alpha$.

3. (Convergence of the costs) Let $Q,V,R,F$ be cost data for this system, and let $J[\alpha^{(n)}]$ be the cost of $\alpha^{(n)}$. Then the real sequence $\bigl(J[\alpha^{(n)}]\bigr)_{n}$ has a limit; this limit is the same for every approximating sequence of $\alpha$; and if $\alpha$ is itself admissible, the limit equals $J[\alpha]$.