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(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+z²

(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+z²

Advanced Engineering Mathematics
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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plz solve question 4(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.
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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