Abstract:
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.
