Let $T>0$ fixed. We consider the optimal control problem for analytic affine systems~: $\ds{\dot{x}=f_0(x)+\sum_{i=1}^m u_if_i(x)}$, with a cost of the form~: $\ds{C(u)=\int_0^T \sum_{i=1}^m u_i^2(t)dt}$. For this kind of systems we prove that if there are no minimizing abnormal extremals then the value function $S$ is subanalytic. Secondly we prove that if there exists an abnormal minimizer of corank 1 then the set of end-points of minimizers at cost fixed is tangent to a given hyperplane. We illustrate this situation in sub-Riemannian geometry.
Publié le : 2000-07-05
Classification:
optimal control,
value function,
abnormal minimizers,
subanalyticity,
sub-Riemannian geometry,
[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
@article{hal-00086284,
author = {Tr\'elat, Emmanuel},
title = {Some properties of the value function and its level sets for affine control systems with quadratic cost},
journal = {HAL},
volume = {2000},
number = {0},
year = {2000},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-00086284}
}
Trélat, Emmanuel. Some properties of the value function and its level sets for affine control systems with quadratic cost. HAL, Tome 2000 (2000) no. 0, . http://gdmltest.u-ga.fr/item/hal-00086284/