[MUSIC] Given a CPS in terms of differential equations or inclusions with constraints and difference equations or inclusively constraints, now we can define a execution or solution to it. A CPS, Given by x in F(x), x+ in G(x) subject to constraints. And notice that for the time being, we are not including inputs. We'll add that, shortly. A hybrid arc, Phi, Is an execution, If we have the following properties. The first property is that the initial value of that hybrid arc is compatible with where the system is allowed to evolve. The initial value is, Phi at 0, t and 0, j. And that needs to belong to C or D. If it isn't D, suddenly, there is a possibility of jumping through this difference inclusion. While if it isn't C, depends whether the flow is pointed in the right direction, or not. In general, this set C might be not closed, but you might have a possibility of starting at the boundary of this set that has not belong to the set and flowing into the set. In which case, we can capture that by closing the set by putting a closure on it. So if we have an initial condition that is compatible, then it already qualifies as a potential execution. The other property that we need to have, is that during flows, we have that the dynamics given by the differential inclusion and the constraint are being satisfied. So we will say that for each j, such that this interval Ij, which was defined already as all t such that the given t, j belong to it, We have. First, that the variation of our time of phi, ordinary time, belongs to the set of possible derivatives or velocities. Or if it is an equation, this will be saying that we have a differential equation solution due in the flow interval. And this will be for almost, All t in Ij. In particular, because we allow ourselves to not require a derivative on the boundary points of these intervals and potentially on sets of measure zero within this interval that could be triggered because non-differentiability values of phi. Furthermore, we would like to have that phi, for every t in this interval, belongs to C. And we're actually going to relax this, This lightly, like asking for this to happen, only on the interior of Ij. So this denotes the interior of this interval, which is essentially equal to (tj, tj + 1) open, when Ij for the given j is given by an interval from (tj to tj + 1) closed. In other words, at the initial and at the endpoint of the flow, we don't really care whether it's in C or not. But in between the evolution over the then set, we have to have the property of being in the flow set. The next property is correspondent to the jumps. The jumps occur whenever there is a t and a j, such that for that same t, j is incremented by one, belongs also to the domain of that hybrid arc. That's the definition of a hybrid arc, and the notion you will get for jumps or the situation you will get for jumps. So for each (t, j) in domain of phi, such that (t, j + 1) is also in the domain of phi, but we will definitely have a jump. And in that situation, the new value of phi, which is phi at (t, j + 1) should be an element of where the state is allowed to jump to when the jump map is evaluated at the jump value of the solution. And moreover, we need to make sure that phi at (t, j) is an element of D which is were the jumps are allowed. These conditions will imply that when satisfied, that hybrid arc is an execution to the system. That execution is no more than a solution to the system, a trajectory to the system, from that usual knowledge of continuous time and discrete time systems that you can think of. In particular if you have a continuous time system, you don't have any jumps. This is essentially what it boil down to the case of a differential inclusion with constraints. But if you further remove the constraint and put here equality, this is no more than the solution to a differential equation with a potentially not a smooth right-hand side f. And when I say smooth, they typically have that. This happens for all t. And similar arguments can be made for the jump cases of the system. So this is notion of execution. This is how we will analytically check a candid execution. Keep in mind that most times, we don't need to check for this. We don't need to compute them ourselves. We either have a numerical solver that will do that for us, or if we have an analytical tool that will not require solving for the solutions, but it will say something about the solutions of the system. We will not even worry about actually defining them and computing them explicitly. [MUSIC]
