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Consider the autonomous system for z € R": ż(t) = f(r(t)) (1) Let r>0 be some fixed constant, and let D = {r: |2|

Consider the autonomous system for z € R": ż(t) = f(r(t)) (1) Let r>0 be some fixed constant, and let D = {r: |2|

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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Consider the autonomous system for z € R": i(t) = f(r(t)) (1) Let r>0 be some fixed constant, and let D = {x: |r| <r}. %3! Suppose there exists a continuously differentiable function V, defined on D such that Vi(0) = 0 and V is positive definite along solutions of Equation (1). Furthermore, suppose for any e E (0, r), Vị itself is not negative semidefinite on {r : |r| < e}. Prove that the origin is unstable. %3D
Transcribed Image Text:Consider the autonomous system for z € R": i(t) = f(r(t)) (1) Let r>0 be some fixed constant, and let D = {x: |r| <r}. %3! Suppose there exists a continuously differentiable function V, defined on D such that Vi(0) = 0 and V is positive definite along solutions of Equation (1). Furthermore, suppose for any e E (0, r), Vị itself is not negative semidefinite on {r : |r| < e}. Prove that the origin is unstable. %3D
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