if the following properties hold.

And we're going to have three properties here.

We're going to write two here and then the other one here.

The first property is that the initial value of the hybrid arc

belongs to the original for operation.

The system can evolve in C or D.

We will guarantee that

these initial value of that function is where I can flow or where I can jump.

Now at times, this set C is not necessarily

a close set and we would like to allow the trajectory to flow into the flow set,

in which case, we will be able to add there a bar and close the set.

Every time that there is an interval of flow,

like I say we find it here,

what I would like to say is that this interval which I define

here as i to the j having known empty interior or non-zero length,

I will have the property that on these I_j's with such a property,

I will have the derivative of the function of phi with respect to

time belongs to F and t,jl.

I'm more aware that I will have that its value belongs to

C. That will be the condition that captures

this property that I have

essentially a solution to the constrained differential inclusion.

And as we describe here for these elements t and j,

such that I have a jump so therefore,

t and t,j plus one is also an element in the domain,

I will need to have the property that if t and j,

and t,j plus one in the domain of phi,

we have that whenever we jump,

the new value belongs

to the image of a jump map evaluated when I jump and suddenly,

this value needs to belong to D. There are,

basically, three conditions that

these arcs that are hybrid will need to satisfy in order to have a solution.

Those conditions are written down here informally.

There are subtleties about whether

these should hold on the open or on the close interval,

and you can look at the references for that with those details.

But the main point is that every time that there is an interval of flow,

we should be able to take a derivative with respect to time and satisfy

the differential constrain with the inclusion or the equation.

And every time that there is a jump,

we should be able to guarantee that the value of the solution after the jump is given by

an element of the solution where the jump map is evaluated at,

and whenever the jump occurs, where in the jump set.

Now, we can consider different cases of CPS,

and I will provide one shortly,

where you can actually compute specifically and

analytically the solution to a system and then write down these arcs.

In many other cases,

you can't do that but you can have a generator of these executions numerically that will

actually give you the approximated execution

to your system from a given initial condition.