Conditional viability for impulse differential games

We introduce the concept of "impulse evolutionary game" and study the concept of "conditional viability" of a subset describing the constraints with a (possibly empty) target under such an impulse evolutionary game. Examples of evolutionary games are usual differential games, differentiable games with history (path-dependent differential games), mutational differential games, etc. Impulse evolutionary systems and games cover in particular "hybrid systems" as well as "qualitative systems". The conditional viability kernal of a constrained set (with a target) is the set of initial states such that for all strategies played by the second player, there exists a strategy of the first player such that the associated run starting from this initial state satisfies the the constraints until it hits the target. The main objective of the paper is to characterize it in several ways.