Some Remarks on Control Strategies for Continuous Gradient Play Dynamics

This work is motivated by the previous studies of approximating the derivative actions on continuous gradient play (GP) dynamics. The novelty lies on the implementation of derivative actions by small time delays. The stability/instability behavior of the GP dynamics with respect to the spectrum of its characteristic matrix is rendered and the corresponding phenomena are explained (well- and ill-posed approximation). The development is represented via a simple geometric approach that surfaces from the analysis. It is proven that sufficiently small delays do not necessarily guarantee the stability of GP dynamics, but by tuning some parameters one may eliminate such instabilities. Illustrative examples complete the paper