[MUSIC] With a notion of an execution for a system without inputs, now we can consider what would be needed in the case that an input is present. In this case, we're going to be looking at a system of the following form. We have some input, we're going to call it gamma. That is not necessarily a sign, it's a generic function of time. Now, we are going to be considering not only the part of the state trajectory or execution, but also the input. Given an input, Gamma, A hybrid arc, phi is such that phi and gamma define an execution input pair if we have the following properties. So now we have the input dependence, also potentially on where I can, it will continuously and where I can evolve discretely. So you can expect that all of the properties that we wrote down will also require not only the state part, which is the but also the input part to also belong to the set C or the set D. Essentially, the first property that we have was about initial condition. And now we have an initial value of the input as well, and now I will like to have that not only to be in C or in D, But also the part using the input B in that set. Keep in mind that now these sets are in a larger dimension because of the input, so this is no more than an extension of what we had before. The one thing that we need to make as an additional assumption is the following. If this input here is also a hybrid signal, a hybrid function and hybrid time domain, then, if the arc that I'm provided with decides to jump at a different instant than the input that I'm provided with, I will need to reconstruct, essentially, these functions so that they match where they are defined. The reconstruction is possible but it's a little bit troublesome. So we will impose that the domain of phi and the domain of gamma coincide. Now, what typically happens is that typically, Gamma is just a function of time. Maybe of some dimension m, which is the input space, let's say. So given phi, which is a hybrid arc, Gamma can be rewritten On the domain of phi, by basically generating a hybrid time domain that is equal to the hybrid time domain of phi. And whenever the hybrid time domain has an interval of flow then you will have a flow of gamma. When every it has jump, there is a new value of j that is incremented by 1, and the value of gamma is the same as before the jump, so it's a trivial jump. So you can always extend this and create a function gamma that is in the right hybrid time domain, but if not, you can massage them and make them match. In that situation, then we will have that for each, j such that ij is not empty, The interior of ij is not empty, But we'll have now the two properties that we had before, the derivative with respect to time to be now belonging to not only the state part, but also the input part, and this will be, as we did before for almost all, t in Ij, and that now not only the state part, but also the input part belong to C for all t in the interior of Ij. And whenever we jump, we'll have essentially a similar structure. So for all Tj in the domain of phi such that (t, j + 1) is also in the domain phi. We will have the property that phi(t,j) and gamma(t,j) belong to D, and the new value of phi after the jump is given by the evaluation of the jump map, At the value at the jump of the state and of the input. That's how we can extend the notion. The only added property is this one here. Again, you want to typically rewrite those inputs. And sometimes we're given inputs that are not compatible to the system, and that's just saying that we should pick a different input. This defines the execution with inputs for this class of dynamical systems modeled in CPS, and now that we have these notions, we can go and say what kind of executions we can have for our system. And, we can actually determine those by studying the properties of the domain of the execution. [MUSIC]