Q9E Consider a systemdx鈫抎t=Ax鈫捖�... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 9: Q9E (page 437)

Consider a system $\frac{d \overset{鈫�}{x}}{d t} = A \overset{鈫�}{x}$ where A is a $2 脳 2$ matrix with $tr A < 0$ . We are told that A has no real eigenvalue. What can you say about the stability of the system

Short Answer

Expert verified

The given system $\frac{d \overset{鈫�}{x}}{d t} = A \overset{鈫�}{x}$ is stable.

Step by step solution

Step 1:Explanation for the continuous dynamical systems with eigen values p±iq

Consider the linear system $\frac{d \overset{鈫�}{x}}{d t} = A \overset{鈫�}{x}$ , where A is the real $2 脳 2$ matrix with complex eigenvalues $p 卤 i q$ and $q 鈮� 0$

Consider an eigenvector $\overset{鈫�}{v} + i \overset{鈫�}{w}$ with eigenvalue $p 卤 i q$ . Then:

$\overset{鈫�}{x} (t) = e^{p t} S (\begin{matrix} \cos (q t) & - \sin (q t) \\ \sin (q t) & \cos (q t) \end{matrix}) S^{- 1} {\overset{鈫�}{x}}_{0}$ , where $S = [\overset{鈫�}{w} 鈥� 鈥� 鈥� \overset{鈫�}{v}]$

Recall that $S^{- 1} {\overset{鈫�}{x}}_{0}$ is the coordinate vector of $x_{0}$ with respect to the basis $\overset{鈫�}{w}, \overset{鈫�}{v}$ .

Explanation of the stability of a continuous dynamical system

For a system, $\frac{d \overset{鈫�}{x}}{d t} = A \overset{鈫�}{x}$ where A is the matrix form.

The zero state is an asymptotically stable equilibrium solution if and only if the real parts of all eigen values of A are negative.

Solution for the stability of the system

Consider A has no real eigen value, which means the real part of all the eigen values are negative.

Thus, the system $\frac{d \overset{鈫�}{x}}{d t} = A \overset{鈫�}{x}$ is stable

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91影视

Consider a system $\frac{d \overset{鈫�}{x}}{d t} = A \overset{鈫�}{x}$ where A is a $2 脳 2$ matrix with $tr A < 0$ . We are told that A has no real eigenvalue. What can you say about the stability of the system

Short Answer

Step by step solution

Step 1:Explanation for the continuous dynamical systems with eigen values p±iq

Explanation of the stability of a continuous dynamical system

Solution for the stability of the system

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91影视

Consider a systemdx鈫�dt=Ax鈫�where A is a2脳2matrix withtrA<0. We are told that A has no real eigenvalue. What can you say about the stability of the system

Short Answer

Step by step solution

Step 1:Explanation for the continuous dynamical systems with eigen values p±iq

Explanation of the stability of a continuous dynamical system

Solution for the stability of the system

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Discrete Mathematics

Decision Maths

Theoretical and Mathematical Physics

Statistics

Calculus

Study anywhere. Anytime. Across all devices.

Consider a system $\frac{d \overset{鈫�}{x}}{d t} = A \overset{鈫�}{x}$ where A is a $2 脳 2$ matrix with $tr A < 0$ . We are told that A has no real eigenvalue. What can you say about the stability of the system