Solving IVPs with Heaviside function where switch depends on function value

Question

I am looking to solve an IVP of the format

$y'(t) = \begin{cases} a, & y(t) < y_0 \\ b, & y_0 \le y(t) < y_1 \\ c, & y(t) \ge y_1 \\ \end{cases}$

That is, the rate depends on the actual function value and switches stepwise between different constant values.

I have looked at using Laplace transforms (it's been a while!) but I am unsure how to use them when the switching is done using the function value and not as a value of $t$.

(for practical context, this is being used to control the feeding and consumption of bottles on a packing line)

Answer 1

That definition divides the plane into three parts:

• $y < y_0$ (below green line in image below)
• $y_0 \le y < y_1$
• $ y \ge y_1$ (above pink line in image below)

For each part slopes $a$, $b$ and $c$ are required.

Here is a sketch of the situation for $a=1/2, b=2, c=1$:

For this situation it is possible to pick a continous $y(t)$, if one point $(t^*, y(t^*))$ is provided.