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The question I have is similar to the one posed in this post with a slight variation. I have a PDE of the form:

$u_t\left(x,t\right)+f\left(t\right)u_x\left(x,t\right)+au\left(x,t\right)=0$

where $f\left(t\right)$ is a control variable and $a$ is a constant. For use in the context of a larger control project, I would like to linearize this expression around some known steady state solution ($u_s$) and control value or setting ($f_s$).

Most of what I have found on PDE linearization so far pertains to cases like the example offered in the answer of this post where the non-linearity arises as some non-linear expression of the $u$ function being solved for. That's also the case in the linearization of an operator post, but instead of looking at the linearization of $u^2$ they look at the linearization of $u_x^2$. In both situations, the solution was to take some epsilon perturbation from the steady state $u=u_s+\epsilon v$, substitute back into the original expression, then eliminate order $\epsilon^2$ or higher terms.

My case seems to be different in the sense that I want to linearize a quantity that depends on terms other than $u$ itself, so I'm unsure if this presents any unique issues that I should be concerned about or if I can just proceed as the other posts suggest. I've been looking for references in the literature for situations like this, but haven't found anything that deals with instances like this yet, so if you can shed any light on this or direct me to a resource that could, that'd be greatly appreciated.

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Leif Ericson
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The PDE $u_t + f u_x = -au$ is linear, even if $f$ depends on $t$. Introducing the variable $v = e^{at} u$, we have \begin{aligned} &v_t = e^{at} u_t + a e^{at} u = -e^{at}(f u_x + au) + a e^{at} u = -fv_x . \end{aligned} According to the method of characteristics (see e.g. this post), solutions of the time-dependent linear advection equation $v_t + f v_x = 0$ can be written $$ v(x,t) = F\left(x - \int_0^t f(s) \,\text ds\right) $$ from which one deduces $u = e^{-at} v$.

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