Controlled State of an Extended Admissible Control

Consider a linear-Gaussian state-observation model on $[0,T]$ with state $X$, a control dimension $k\ge1$, a control matrix assignment $B$ as in Controlled State and Controlled Observations in the Linear-Gaussian Model, and an extended admissible control $\alpha$ with values in $\mathbb{R}^{k}$.

The controlled state of $\alpha$ is the family $X^{\alpha}=(X^{\alpha}_t)_{t\in[0,T]}$ of tuples defined componentwise by

where $c^{\alpha}$ is a fixed choice of a family satisfying claims 2(a) and 2(b) of Convergence of Corrections and Costs along Approximating Sequences of an Extended Admissible Control for a chosen approximating sequence of $\alpha$.

Well-definedness. Such a family $c^{\alpha}$ exists by claim 2 of Convergence of Corrections and Costs along Approximating Sequences of an Extended Admissible Control, and by claim 2(c) there any two choices (for any approximating sequences) agree almost surely at every $t$, so $X^{\alpha}_t$ is determined up to almost sure equality at every $t$.

Consistency. If $\alpha$ is an admissible control, then by claim 2(c) of Convergence of Corrections and Costs along Approximating Sequences of an Extended Admissible Control and claim 2 of Superposition Decomposition of the Controlled State and Observations the family $X^{\alpha}$ agrees almost surely at every $t$ with the controlled state already defined for admissible controls, so the same symbol may be used for both.