I am not an expert in control theory, so I don't quite understand the Filippov set-valued maps conditions, but I do have a control system $\dot x(t) = F[x(t)]$ where $F[x(t)]$ is a scaled subdifferential of a convex (nonsmooth) function. Is that sufficient to say my system is Filippov?
Edit: Some definitions:
The subdifferential of a convex function at $x$ is defined as the set $\partial f(x)$ where $$ f(x) - f(y) \geq g^T(x-y), \quad \forall g\in \partial f(x). $$
Probably the one quality I would think is useful for this question is that subdifferentials are maximally monotone, in that for all $x$ and $\bar x$, $g\in \partial f(x)$ and $\bar g \in \partial f(\bar x)$ if and only if $$ (g-\bar g)^T(x-\bar x)\geq 0. $$
Visually, these sets $\partial f(x)$ are always convex, and has some kind of continuity w.r.t. $x$, in that if we define the distance between sets as $$ D(S,T) = \min_{x\in S,y\in T}\|x-y\|_2,$$ then $D(\partial f(x),\partial f(x+\Delta))$ is a continuous function of $\Delta$. (I'm like 85% sure of this last bullet, not 100%.)
A Filippov representable set is kind of like an intersection of maps $$ F_f(x) := \cap_{L(N)=0}\cap_{\delta > 0} \overline{\text{co}} f((B_\delta(x))\setminus N) $$ where $L(N)=0$ means $N$ is a set of Lebesgue measure 0.
This is also a paper that I am heavily relying on: https://arxiv.org/pdf/2003.00436.pdf. It would appear from this paper that subdifferentials are a special case of Filippov representable sets, but I can't be 100% sure I'm intepreting it correctly since the language is quite esoteric. It almost feels like this continuity requirement of subdifferentials is relaxed to be allowed to break over a set of measure 0. I am wondering if anyone can confirm?
Edit 2: where this question is coming from.
Ultimately what I am trying to do is to understand the flow mechamism behind a method for convex optimization, which can be framed as $$\dot x(t) = c(t) \partial \sigma(x(t))$$ where $c(t)$ is some decaying nonnegative scalar quantity, and $\sigma(x)$ is the support function of $x$ over some convex set. $\partial \sigma(x)$ is the subdifferential of $\sigma$. In general, $\sigma(x)$ is highly nonsmooth, but it is a closed convex function, so it should have nice properties.
In terms of the flow mechanism, from what I understand, if I can characterize $\dot x = c(t) \partial \sigma x(t)$ as a Filippov inclusion system, then existence of at least one convergent trajectory (convergent in the sense of discretization) is assured. This makes the trajectory I am trying to analyze justified. Of course, if there are other types of conditions that are also possible, say, if this qualifies as a Krasnowskii system and thus convergence and existence is guaranteed, then that serves my needs too.
I would say that the right way to look at your system is in terms of the Krasovskii regularization. In that case, your system will be a differential inclusion with a map that is single-valued at differentiable points and set-valued (e.g. box-valued) at non-differentiable ones.
The issue with Filippov is that, from the description of your problem, it seems that non-differentiability of $\sigma$ only occurs at countable number of points. However, Filippov ignores the behavior of the map on a set of measure zero, which is problematic here as non-differentiability only occurs on measure zero sets. In this regard, the Filippov regularization will simply ignore all the points where the function is not-differentiable.
More details in http://www.numdam.org/item/10.1051/cocv:2008008.pdf and the references therein.