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) = 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)→ ∞o as lu] → ∞o, then all solutions are bounded for t > 0; (c) equilibrium points have the form (u*, 0) where u* is a root of g; (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) = 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)→ ∞o as lu] → ∞o, then all solutions are bounded for t > 0; (c) equilibrium points have the form (u*, 0) where u* is a root of g; (1)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
6
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) = ½v² +G(u) is a Lyapunov function for (1);
(b) if G(u)→ ∞o as u → ∞o, 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.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 5 steps
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
