Variational analysis and sensitivity relations of the bilateral minimal time functions for nonlinear differential inclusions. (c2019) - Chamoun, Samara Sarkis (2019)
The main objective of this thesis is to study the '0-convexity of the epigraph of the bilateral minimal time function for a nonlinear control system. There are three parts in our work. In the first part, we study the variational analysis of the bilateral minimal time function under the Standing Hypotheses. One of the main results of this part is a relation between the proximal normal cones to sub-level sets of the bilateral minimal time function and its epigraph. The second part is devoted to the generation of sensitivity relations for the bilateral minimal time function. More precisely, we prove some propagation results for the proximal (horizontal) subdifferential along optimal trajectories. In the third part, we use the results of the first two parts to study the regularity of the bilateral minimal time function for a nonlinear control system. Among other assumptions, we prove that the continuity of the bilateral minimal time function near a point is suffcient for the '0-convexity of its epigraph near this point. This extends, to the nonlinear case, a similar result proved by Nour in 2013 for a linear control system.