Can someone please clarify what the difference (no pun intended) between the two is?
I am reading this tutorial and at the very start they state that a differential inclusion is a solution to
$
\frac{\mathrm{d}}{\mathrm{d}t}x(t) \in F(t, x(t))
$
So that means that the derivative of $x(t)$ is included in $F$ which is a function the argument $t$ and the $x(t)$ itself. Then a solution is a family of functions, right? But why is this different the just a differential equation? I suppose it's meant to be more general, but right now I'm failing to see where.
P.S. no tag for differential inclusions, so differential equations it is.
Best Answer
An ordinary differential equation says what the derivative must be, in terms of the function itself and its variable.
An ordinary differential inclusion says the derivative must lie in a specified set, which may also depend on the function and independent variable.
So, if the set always consists of one point, the inclusion is in fact an equation.
The distinction is blurred if one allows implicit differential equations, in which the derivative is not isolated. These can be interpreted as inclusions, since the derivative is constrained to a set. Conversely, every inclusion can be written as an implicit differential equation by using the indicator function of the set on the right hand side of the inclusion.
But this notational juggling is pretty pointless: as you keep reading beyond the first pages of the survey, you'll see that the theory of differential inclusions is quite a bit different in its goals and methods.
For comparison: every graph can be thought of as a matrix, but this does not mean we want to do that, nor does it make graph theory a part of linear algebra.
A solution is a function, in either case.