[MUSIC] So with a notion of execution for a CPS given by differential equation and difference equations with constraints and maybe inclusions. We will like now to analyze their behavior and one very powerful way to analyze it is to use simulation, essentially approximate numerically the behavior of a system by computing the executions over time. The idea here is that we would like to have a modular way to do the simulations. So we would like to now go back to our potentially specific structure of a CPS where here we have the physics. And here we have the cyber and maybe these are interconnected in a figurative fashion using some, Interfaces. As we know, each of these blocks, four specific choices of the interfaces of the cyber. The physics can be modern mathematically, and this can be broken down as a CPS with a state x, potentially some impute gamma and a flow map, and a flow set and a gem up on a jump set. And this at the overall description this at the final overall description at the system. Now what I would like to have is that given an initial state and an initial input and also the input to apply, We will like to determine Phi, which is defined as the execution of the system. Now whether we care about this or we would like to have potential analysis from different initial conditions for many purposes, in particular, In-variance, Attractivity, Stability Logic formula satisfaction, in particular, temporal logic, Analysis. So the way we can do this is by, in a ideal situation, building individual blocks into our tool that will generate approximations of these. And that tool will allow us to interconnect this blocks, it will allow us to define what the initial values of the trajectory of the input, and the input itself would be, and propagate forward in time over a finite horizon. So using simulation, Provides An approximation, Of phi, Over a habit time window Of the form 0 to capital T across 0, 1 to all the way to capital J where these T and J are finite. In other words, numerically we can envision that it is possible to simulate the system or the finite window. And hopefully those simulations can tell us something representative of the region system in order words these approximation is very good approximation of the true execution. And maybe again we would like to use that information to get an idea about invariance properties, attractivity, and stability, and maybe whether that formula is being satisfied. All that type of analysis for these type of systems a first cut step. A first iteration could be to simulate the system and understand the behaviors, in particular because the systems are complex. And throw into it analytical tool directly might not be feasible without some specification of the dynamics. [MUSIC]