Necessary conditions for stability and attractivity of continuous systems

Considered in this paper are control systems of the form x· = f(x,u). For such systems a number of related necessary conditions for various forms of stability and attractivity are presented. The paper starts by showing that Brockettʹs necessary condition for stabilizability via smooth feedback still persists if f is continuous and the class of allowable u increased to include continuous feedbacks. Using similar ideas to those used to prove the continuous Brockett result, again only assuming continuity of f and u, necessary conditions are then derived for global attractivity and for ultimate boundedness.