[SOUND] Giving out execution to the system we can now determine some properties and classify them. So the first type of classification is what we will call nontrivial. Let's consider just an execution that does not have an input component but if it has an input the same ideas apply. So given a phi, which is an execution, we will define the following notions. The first one will be what we call nontrivial. And it will be nontrivial if The domain where they are defined has at least, Two points. If you think about this, if phi initial is in c or in d, if it is in d, then there will be a jump, possible for that phi. And if that phi choose jumps, then suddenly you will have 0,0, t0, j = 0 and t0, j = 1 in the domain of phi. If you have a phi that initially is only in c but not in d, in order to have two points in the domain, you will need to have a little bit of flow. And a little bit of flow within c will actually generate more than two points if we generate infinitely many points. But in any case, it will be a nontrivial execution for that case. One other specific case is continues, By continues we mean the the domain of phi is such that no jumps are possible. Therefore, this domain is a subset of non-negative reals cross 0, no jumps. And you can relate to these by ploting the domain. I'm noticing the domain will only be contained for j equal to 0. Maybe up to a point, maybe all the way to infinity, it doesn't matter too much. Similarly, you can define a dual of this which will be discret. If, The domain of phi is contained in zero for t cross zero one, two, and so on. Okay, so you can also think about the situation where now you have a [INAUDIBLE] time domain that is only existing in the t equals zero and a j possible. This would be my 0 here, and then this will be my 1, and then 2, and 3. And let's say that this is the domain that you have. [COUGH] The next type of solution that I would like to define is what we call complete, That execution or solution, is complete if the domain of phi is unbounded. So when you were to plot it, it will need to be in bounded either in the t-direction or in the j-direction What this allows us to do, in particular, is to then determine where the value of the execution will go, as either t or either j goes to infinity. And that's a typical property of interest when you study stability of a system, attractivity of a set, and so on. The next type of notion that we would like to add, which is a very specific notion for this type of systems because they combine continuous and discrete, is a notion of zeno. And this will be the case when the execution is complete and, The domain of phi is bounded, In the t-direction. So in this situation you will have, A typical [INAUDIBLE] domain of the following form. Imagine that I have a timer that every time that it flows, you have a reset. Such that the next time we have a flow, it will flow for half the amount of time. So I'm trying to describe this here, so every time the amount of time is cut in half, that you flow compared to the previous time that flow curve. It turns out that these series of intervals define the domain of this execution that is Zeno, because it's complete and because it's bounded in the t direction. It will need to go to infinity on J and actually because of the there will be some time call it tz for zenith time where these segments will converge to. This appears in mechanical system, with impulse like post bouncing on the ground, and it's actually making the solution to have only a limit, as j goes to infinity but it turns out that t will actually go to final value. So you have infinitely many events over a finite number of a finite amount of flow, okay? And whether the limit of exertion exists or not is depending on the system, so it's not general. But for instance, for these Is system of a bouncing ball, we actually have that the energy of the ball goes to zero. And that the limit of the solution is actually for the position is zero, and for the velocity is zero, as j goes to infinity and as t goes to this particular Zeno time. So you can actually determine where things converge in line of presence of this behavior called zero. The last type of notion that we will like to define here is what is called maximal. This will correspond to the property that maximal if phi cannot be further extended, By either, Flow or jump. In other words, what we can actually write down formally is that there is now another execution, call it psi, such that psi is equal to phi, For all t and j in the domain of phi, and the domain of psi minus the domain of phi is not empty. So this is empty symbol which shouldn't be confused with a phi. In other words you are looking at the execution for which you have exhausted your capability of flowing and jumping. And those maximum solution will represent the behavior of a system fully to some extent. Now, if the maximum solutions are complete, then certainly you will not be able to continue forward, and you have actually the capability of taking limit as either t or j goes to infinity. Depending on the structure of the domain of that particular solution. However, if this maximum solution is not complete then there's typically an issue in the system or in the model of the system that actually corresponds to having a premature ending of the solution. That should be analyzed, it could be that it's purposely there for some modeling needs or it's just a failure in analyzing the system. Typically when you have a continuous time system or a discrete time system, you put enough properties on the generators the flow map and the gen map. So that every maximum solution is complete and you don't need to worry about a premature stopping or not be able to continue further in time. Like for instance in continues to insist that you may want to prevent final escape time which will make maximum solutions all being complete when the flow map is regular enough in particularly if it's locally [INAUDIBLE] you will have completeness. In general for hybrid systems, we might have a situation where the state of the [INAUDIBLE] map out size here indeed so we might have a premature stop in. In other words maximum solutions are not complete or that the execution reaches a point in the flow set where the floor map is pointing upward. And therefore if that point is not also in the jump set then there is no possibility of continuing forward in time. So you are pretty much stuck as a solution at that particular point. [MUSIC]
