Admissible Control for the Linear-Gaussian State-Observation Model

Consider a linear-Gaussian state-observation model on $[0,T]$, with notation and fixed versions as there, and let $k\ge1$ be a natural number.

An admissible control with values in $\mathbb{R}^{k}$ for the model is a family $\alpha=(\alpha_t)_{t\in[0,T]}$, where each $\alpha_t=(\alpha^{1}_t,\dots,\alpha^{k}_t)$ is a tuple of square-integrable random variables on $(\Omega,\mathcal{F},P)$, such that:

(i) for each $\kappa\in\{1,\dots,k\}$ the component family $(\alpha^{\kappa}_t)_{t\in[0,T]}$ is mean-square continuous;

(ii) for every $t\in[0,T]$ and every $\kappa\in\{1,\dots,k\}$, the random variable $\alpha^{\kappa}_t$ is almost surely equal to a $\mathcal{G}_t$-measurable square-integrable random variable, where $\mathcal{G}_t$ is the observation $\sigma$-algebra of the model — that is, of the observation process $u$ of the model itself, without control — and measurability means that preimages of Borel sets belong to $\mathcal{G}_t$.