AIMS Mathematics (Sep 2024)
On the optimal second order decrease rate for nonlinear and symmetric control systems
Abstract
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.
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