With a hybrid time, now we can provide a notion of execution. All we need to do is to define the value of the functions. This is what we called Hybrid Arc. As we did in a previous video, an execution to a CPS will be given by a generic function that has a domain with a structure of a hybrid time domain. And now, we're going to add the following property, hybrid arc. What does this mean? This means that first and foremost, the domain of the function is a hybrid time domain. So every time that one picks an element on the domain, one can write that as the union of close intervals index by j, and the function defined by fixing j is such that it is differentiable at almost every point t in the following interval. We're going to collect all the t's such that for the given j, (t,j) belongs to the domain. And when we say almost every t, we are referring to every point except those in a set of measure zero, where the measure is using the Lebesgue measure. In particular, these allow us to say that if we have an interval of flow and then a jump, then on the boundaries of that interval, we don't need to worry about taking the derivatives. And moreover, if the flow map which is the object that defines the continuous change of these executions is non-smooth, we might have in particular points of non-differentiability of those executions, and we don't need to worry about differentiating the executions at those points. We can think again about the problem of the timer. Now, we can draw in a 3D plot the state of the timer problem I was referring in a previous video. The idea is essentially to have a timer. So x is a timer that flows as time evolves to x = 1, and then resets to zero. So this appears in many CPS, and in that case, we can now draw here. If we initialize a timer at 0, and this is 1, we can now see that this will be the initial evolution of the timer. Time from zero. It will evolve all the way to one. When that occurs, it's going to be reset to 2. The J will be reset, sorry, to 1. The x will be reset to zero. So it will be this point here. From where is going to have another period of flow of one unit according to this model. And the timer itself will flow according to these linear evolution with time. After another one second occurs, now is going to be the case that j gets reset to 2. And then we're going to re-flowing from t equal to 2, all the way to 3. And so this continues. So the hybrid arc here is given by this function which is a function of t and j is differentiable in the open zero to one which equal to zero. It's derivative is equal to one is differentiable on the open one to two where j equal one, and the derivative is equal to one, and so on. So it satisfies this property of being differentiability at almost every point. In particular, these property is satisfied when the, in general, this function is locally, absolutely, continuous. Again, for every j such that t and j is in the domain, and this will only make sense whenever you have up to unit flow. Otherwise, you will be satisfied. Arbitrarily, if you just skip jumping because a point is only absolutely continues come up with respect to t. This defines the hybrid arc, but we're now left to do is to define properties on this hybrid arc, so that it qualifies for a solution or an execution to the CPS given by differential equations and encryptions with constraints and different equations or inclusions with constraints.
