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c) Is this equilibrium point asymptotically stable? Explain why.

c) Is this equilibrium point asymptotically stable? Explain why.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 17EQ
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PART C

Problem 2. Consider the following linear system *(t) -2.x(t) + y(t) ý(t) x(t) – y(t) a) Show that the "energy" E(x, y) the multivariable chain rule verify that x² +y? is non-increasing along the solution, i.e., using d E(x(t), y(t)) < 0 dt b) Use part a) to show that the origin is stable, i.e., suppose that you are given R > 0, then provide a sufficient condition on r > 0 such that, if x² (0) +y²(0) < r², then we can conclude that the solution x(t), y(t) satisifes x?(t) + y? (t) < R² for all t > 0. [Argue as done in class for the simple harmonic oscillator system] c) Is this equilibrium point asymptotically stable? Explain why.
Transcribed Image Text:Problem 2. Consider the following linear system *(t) -2.x(t) + y(t) ý(t) x(t) – y(t) a) Show that the "energy" E(x, y) the multivariable chain rule verify that x² +y? is non-increasing along the solution, i.e., using d E(x(t), y(t)) < 0 dt b) Use part a) to show that the origin is stable, i.e., suppose that you are given R > 0, then provide a sufficient condition on r > 0 such that, if x² (0) +y²(0) < r², then we can conclude that the solution x(t), y(t) satisifes x?(t) + y? (t) < R² for all t > 0. [Argue as done in class for the simple harmonic oscillator system] c) Is this equilibrium point asymptotically stable? Explain why.
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