Solving non-linear constraint satisfation problems involving time-dependant functions

International audience; In this paper, we consider the resolution of non-linear constraint satisfaction problems where the variables of the systems are trajectories (functions from R to Rn). We introduce the notion of tubes as intervals of functions, for which the lower and upper bounds are trajectories with respect to the inclusion. We then define basic operators and prove propositions verified by tubes. We show the possibility to build contractors on tubes and propagate constraints to solve problems involving time-dependant functions as the unknown variables.We show that the approach is particularly powerful when inter-temporal equations (e.g. delays) are involved. Finally, in order to illustrate the principle and efficiency of the approach, several test cases are provided.