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Do you really mean $Ex_t$? That's the unconditional mean without looking at any of the data, which is a constant. Normally, the Kalman filter tells you how to compute the conditional mean based on the data you have at a particular moment in time.

I'm not familiar with the book, but I assume that you mean what the Wikipedia page calls the predicted state estimate.

You don't say anything about measurement error. Do you observe $x_t$ exactly? By assumption, since you choose $u_t$ you know it at time $t-1$. Then the formula is just telling you that $$ E(x_t | u_t, x_{t-1}) = A E(x_{t-1} | u_t, x_{t-1}) + B E(u_t | u_t, x_{t-1}) + E(\epsilon_t | u_t, x_{t-1}). $$ Since $E(x_{t-1} | u_t, x_{t-1}) = x_{t-1}$ and $E(u_t | u_t, x_{t-1}) = u_t$, and the noise is independent, then $$ E(x_t | u_t, x_{t-1}) = E(x_{t-1}) A x_{t-1} + B u_t. $$

If you don't observe $x_t$ exactly, then you observe it with some error, given by $$ z_t = H_t x_t + \nu_t, $$ where $z_t$ is your observation at time $t$ and $\nu_t$ is again Gaussian white noise, independent of everything else.

Now, at time $t-1$ you only know $z_1, \ldots, z_{t-1}$, so the best you can do is $$ E(x_t | u_t, z_1, \ldots, z_{t-1}), $$ which I'll write as $$ E_{t-1} (x_t), $$ since it's the conditional mean given all information at time $t-1$ (including $u_t$, which you choose at time $t-1$.

Using the definition, $$ E_{t-1}(x_t) = A E_{t-1}(x_{t-1}) + B E_{t-1}(u_t) + E_{t-1}(\epsilon_t), $$ but the most this simplifies is to $$ E_{t-1}(x_t) = A E_{t-1}(x_{t-1}) + B u_t. $$ With measurement error, the Kalman filter by design just works by calculating the two conditional means $E_{t-1}(x_t)$ and $E_{t-1}(x_{t-1})$, and no other conditional or unconditional means.
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Do you really mean $Ex_t$? That's the unconditional mean without looking at any of the data, which is a constant. Normally, the Kalman filter tells you how to compute the conditional mean based on the data you have at a particular moment in time.

I'm not familiar with the book, but I assume that you mean what the Wikipedia page calls the predicted state estimate.

You don't say anything about measurement error. Do you observe $x_t$ exactly? By assumption, since you choose $u_t$ you know it at time $t-1$. Then the formula is just telling you that $$ E(x_t | u_t, x_{t-1}) = A E(x_{t-1} | u_t, x_{t-1}) + B E(u_t | u_t, x_{t-1}) + E(\epsilon_t | u_t, x_{t-1}). $$ Since $E(x_{t-1} | u_t, x_{t-1}) = x_{t-1}$ and $E(u_t | u_t, x_{t-1}) = u_t$, and the noise is independent, then $$ E(x_t | u_t, x_{t-1}) = E(x_{t-1}) + B u_t. $$

If you don't observe $x_t$ exactly, then you observe it with some error, given by $$ z_t = H_t x_t + \nu_t, $$ where $z_t$ is your observation at time $t$ and $\nu_t$ is again Gaussian white noise, independent of everything else.

Now, at time $t-1$ you only know $z_1, \ldots, z_{t-1}$, so the best you can do is $$ E(x_t | u_t, z_1, \ldots, z_{t-1}), $$ which I'll write as $$ E_{t-1} (x_t), $$ since it's the conditional mean given all information at time $t-1$ (including $u_t$, which you choose at time $t-1$.

Using the definition, $$ E_{t-1}(x_t) = A E_{t-1}(x_{t-1}) + B E_{t-1}(u_t) + E_{t-1}(\epsilon_t), $$ but the most this simplifies is to $$ E_{t-1}(x_t) = A E_{t-1}(x_{t-1}) + B u_t. $$ With measurement error, the Kalman filter by design just works by calculating the two conditional means $E_{t-1}(x_t)$ and $E_{t-1}(x_{t-1})$, and no other conditional or unconditional means.