(c) Consider the system x'(t) =ay² - x, y' (t) = -y-ax². x' (t)-y-2x, y'(t)=2x-y-x³.
(c) Consider the system x'(t) =ay² - x, y' (t) = -y-ax². x' (t)-y-2x, y'(t)=2x-y-x³.
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
plz solve question 5(a) it within 30-40 mins I'll give you multiple upvote
![I
1
Question 4
(a) Show that V(x, y) = ln(1 + ²) + y² is a Lyapunov function for the system
x'(t)= x(y-1), y(t)=-
1+x²
(b) Let a > 0. Show that V(x, y) x² + 2y2 is a strict Lyapunov function for the
system
x' (t)=ay² -x, y' (t)=-y-ax².
(c) Consider the system
x' (t)-y-2x, y(t)=2x-y-x³.
Using the Lyapunov function V = (x + y)² +¹. show that the origin is Lyapunov
stable.
Question 5
(a) Using the Lyapunov function candidate V(x, y, z)=(x² + y² +22), investigate
stability of the origin of the system
x' (t) = -x + x²z, y(t)=z, 2'(t)=-y-z-x³.
(b) Using the Lyapunov function candidate V(x, y, z) =
stability of the origin of the system
2²+¹, investigate
x' (t)= y, y' (t)-³-y³-2³, 2'(t)=-z+y.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5ddabf74-e977-4c29-a35f-5120fa90316b%2F5c1ea15f-76b3-42a1-93c2-f10d291581b2%2F3rk8o2n_processed.png&w=3840&q=75)
Transcribed Image Text:I
1
Question 4
(a) Show that V(x, y) = ln(1 + ²) + y² is a Lyapunov function for the system
x'(t)= x(y-1), y(t)=-
1+x²
(b) Let a > 0. Show that V(x, y) x² + 2y2 is a strict Lyapunov function for the
system
x' (t)=ay² -x, y' (t)=-y-ax².
(c) Consider the system
x' (t)-y-2x, y(t)=2x-y-x³.
Using the Lyapunov function V = (x + y)² +¹. show that the origin is Lyapunov
stable.
Question 5
(a) Using the Lyapunov function candidate V(x, y, z)=(x² + y² +22), investigate
stability of the origin of the system
x' (t) = -x + x²z, y(t)=z, 2'(t)=-y-z-x³.
(b) Using the Lyapunov function candidate V(x, y, z) =
stability of the origin of the system
2²+¹, investigate
x' (t)= y, y' (t)-³-y³-2³, 2'(t)=-z+y.
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