On the optimal second order decrease rate for nonlinear and symmetric control systems

When a control system has all its vector fields tangent to the level set of a given smooth function $u$ at a point $\hat{x}$, 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 $\hat{x}$ with respect to the vector fields of the system.