Paper 4, Section II, C
Part II, 2011
(i) State and prove Lyapunov's First Theorem, and state (without proof) La Salle's Invariance Principle. Show by example how the latter result can be used to prove asymptotic stability of a fixed point even when a strict Lyapunov function does not exist.
(ii) Consider the system
Show that the origin is asymptotically stable and that the basin of attraction of the origin includes the region .