Q7.6-36E Show that the zero state is a st... [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 7: Q7.6-36E (page 381)

Show that the zero state is a stable equilibrium of the dynamical system $\overset{鈫�}{x} (t + 1) = A \overset{鈫�}{x} (t)$ if (and only if ) $\underset{t 鈫� 鈭�}{l i m} A^{t} = 0$ (meaning that all entries of $A^{t}$ approach zero).

Short Answer

Expert verified

It has been proved that zero state is stable.

Step by step solution

Define stability:

Stability is one of the most basic requirements for numerical models, which extends to mostly linear problems. If every solution that was initially close to it is always close to it. It is said to be stable and asymptotically stable although initially every solution close to it meets t. $鈫� 鈭�$

Prove zero state is stable:

The vectors $e_{1, 鈥�,} e_{n}$ are all distributional vectors, and they also form a basis for ${鈩�}^{n}$ .Then, for any $x 鈭� {鈩�}^{n},$ applies $x = {鈭�}_{i = 1}^{n} a_{i} e_{i}$ ,

$\underset{t 鈫� 鈭�}{l i m} A^{t} x = \underset{t 鈫� 鈭�}{l i m} A^{t} {鈭�}_{i = 0}^{n} a_{i} e_{i}$

$= \underset{t 鈫� 鈭�}{l i m} {鈭�}_{i = 0}^{n} a_{i} A^{t} e_{i} = {鈭�}_{i = 0}^{n} a_{i} \underset{t 鈫� 鈭�}{l i m} A^{t} e_{i} = {鈭�}_{i = 0}^{n} a_{i} 路 0$

$\underset{t 鈫� 鈭�}{l i m} A^{t} = 0 .$ where $x 鈭� {鈩�}^{n}$

$\underset{t 鈫� 鈭�}{l i m} A^{t} x = 0$ which means that the latter equation also applies for distributional vectors, so zero state is stable.

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91影视

Show that the zero state is a stable equilibrium of the dynamical system $\overset{鈫�}{x} (t + 1) = A \overset{鈫�}{x} (t)$ if (and only if ) $\underset{t 鈫� 鈭�}{l i m} A^{t} = 0$ (meaning that all entries of $A^{t}$ approach zero).

Short Answer

Step by step solution

Define stability:

Prove zero state is stable:

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91影视

Show that the zero state is a stable equilibrium of the dynamical system x鈫�(t+1)=Ax鈫�(t)if (and only if )limt鈫�鈭�At=0(meaning that all entries ofAtapproach zero).

Short Answer

Step by step solution

Define stability:

Prove zero state is stable:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

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Study anywhere. Anytime. Across all devices.

Show that the zero state is a stable equilibrium of the dynamical system $\overset{鈫�}{x} (t + 1) = A \overset{鈫�}{x} (t)$ if (and only if ) $\underset{t 鈫� 鈭�}{l i m} A^{t} = 0$ (meaning that all entries of $A^{t}$ approach zero).