@article{Costantini2024, 
author = {Mauro Costantini and Pierpaolo Soravia},
title = {On the optimal second order decrease rate for nonlinear and symmetric control systems},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {10},
pages = {28232-28255},
keywords = {nonlinear mathematical control, controllability conditions, higher order controllability conditions, higher order decrease rate, asymptotic distribution of eigenvalues, control Lyapunov function},
url = {https://www.sciopen.com/article/10.3934/math.20241369},
doi = {10.3934/math.20241369},
abstract = {When a control system has all its vector fields tangent to the level set of a given smooth function  u at a point  x^, under appropriate assumptions that function can still have a negative rate of decrease with respect to the trajectories of the control system in an appropriate sense. In the case when the system is symmetric and  u has a decrease rate of the second order, we characterise this fact and investigate the existence of a best possible rate in the class of piecewise constant controls. The problem turns out to be purely algebraic and depends on the eigenvalues of matrices constructed from a basis matrix whose elements are the second order Lie derivatives of  u at  x^ with respect to the vector fields of the system.}
}