Consider the dynamical system: x = f(x), where x = 0 is the only equilibrium and f is (Lipschitz) continuous. Our goal in this problem is demonstrating that the equilibrium x = 0 is Lyapunov stable if there exists a Lyapunov function V. Steps (a), (b) and (c) are intermediate steps of the proof that lead to (d), the main statement. Note: We expect clear, concise arguments that refer back to the properties of V, but do not expect high level of rigor while doing so. Assume that there exists a continuously differentiable Lyapunov function such that V(0) = 0 and V(x) > 0 for x R" and x0 V(x) 0. You are given > 0. Consider the e-ball: (11) (12) Be = {x R" ||x|| }. (a) Let a = min||||= V(x). Show that a > 0. (5 pts.) (b) Consider the sublevel set of Lyapunov function intersecting Be B = {x: V(x) B} n Be where 3 (0, a). Show that B C Be (sketched in Figure 4. Hint: Assume ng is not in the interior of Be, hence there is a point p on the boundary of Be, i.e. ||p|| = and show that there is a contradiction. (5 pts.) (c) Show that B is positively invariant, i.e. any trajectory starting in Bat t = 0 stays in NB for all t 0. (5 pts.) (d) Since V(x) is continuous and V(0) = 0, we know that there exists a d > 0 such that Bs = {x ||x|| 8} CNB. Observe that B C NB C Be in the Figure 4 For any initial condition x(0), argue that x(0) Bs x(0) NB x(t) = NB x(t) = Int(B) = {x : ||x|| < } for all t 0 using the subset relations and the positive invariance property. Remark: Note that for any e > 0, there exists a > 0 such that the above statement holds. (10 pts.)