I wrote down this equation which is the mathematical model of a system. Is there any way to get $V_c(t)$ in a closed-form expression?
$V_c(t+d) = V_s \cdot U\left(V_r -\frac{V_c(t)}{V_s}\frac{I}{C}t\right)\qquad(1)$
where $V_s,\,V_r,\,I,\,C,\,d$ are real and positive, $t$ is the time and $U(\cdot)$ is the Heaviside function.
Thank you
A
EDIT
I'll try to give more details about the system, if it helps. It's a closed-loop oscillating system whose output is $V_c(t)$:
$V_c(t+d) = V_s \cdot U\left(V_r - V_a(t)\right)\qquad(2)$
and
$ V_a = \frac{V_c(t)}{V_s}\frac{I}{C}t\qquad(3)$
By replacing $(3)$ in $(2)$, I get $(1)$.
Also I forgot to mention that $V_a(0) = 0$, therefore $V_c(t) = 0$ if
$0 \le t \le d$.