Consider the ODE u" + b(u)u' + g(u) = 0, where b and g are continuous functions, and b is positive. In system form, u = v v' = -b(u)v-g(u). Let G(u) = f¹ g(u) du be an antiderivative of g. Show that (a) the function V(u, v) = ½v² +G(u) is a Lyapunov function for (1); (b) if G(u) → ∞ as u → ∞, then all solutions are bounded for t ≥ 0; (c) equilibrium points have the form (u*, 0) where u* is a root of g; (d) all possible w-limit sets are the equilibrium points. (1)
Consider the ODE u" + b(u)u' + g(u) = 0, where b and g are continuous functions, and b is positive. In system form, u = v v' = -b(u)v-g(u). Let G(u) = f¹ g(u) du be an antiderivative of g. Show that (a) the function V(u, v) = ½v² +G(u) is a Lyapunov function for (1); (b) if G(u) → ∞ as u → ∞, then all solutions are bounded for t ≥ 0; (c) equilibrium points have the form (u*, 0) where u* is a root of g; (d) all possible w-limit sets are the equilibrium points. (1)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Consider the ODE u" + b(u)u' + g(u) = 0, where b and g are continuous functions, and b is
positive. In system form,
u' = v
v = -b(u)v – g(u).
(1)
Let G(u) = f" g(u) du be an antiderivative of g. Show that
(a) the function V (u, v) = }v² + G(u) is a Lyapunov function for (1);
(b) if G(u) ∞ as u → x, then all solutions are bounded for t > 0;
(c) equilibrium points have the form (u*, 0) where u* is a root of g;
(d) all possible w-limit sets are the equilibrium points.
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