Consider the dynamical system: x = f(x), where x = 0 is the only equilibrium and...

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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 x‡0 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(B₂) = {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.) 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 x‡0 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(B₂) = {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.)

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a Show that the equilibrium x 0 is invariant under the system Let x 0 be the equilibrium point of th... View the full answer