This is not a specific problem, but a question about the application of a theory. We have these equations of the form $dx/dt=f(x_t)$ where a given point $x$ moves along some path over time.
I am currently facing a modelling problem where $x(t)$ is a sequence of step functions: $x(t)$ takes the value of the constant $c$ in some period $(t_k,t_{k+1})$ and takes another value from $t_{k+1}$ based on the dynamic evolution. I am mainly interested in the stability notions of dynamical systems (Lyapunov, uniform, etc).
The step functions I have in my equation are semi-differentiable only, so it may not be mathematically appropriate to describe them by $dx/dt$. Is there any other way to adapt the usual results involving differential equations and to use them for such $x$?
PS: I do not want to transform to a discrete system, because I am typically interested in the rate of convergence - which depends upon the exponent.