Given the system <mi mathvariant="bold">x &#x2032; </msup> = A <mi mathvarian

ureji1c8r1 2022-05-11 Answered
Given the system
x = A x
A = [ a 0 4 1 1 0 2 a 0 3 ]
In what interval of a is the system asymptotically stable, and for what value of a is the system stable/unstable if such a case even exists?
For a system to be asymptotically stable the real part of all eigenvalues must be negative.
( λ ) < 0 , for all λ
Since we are dealing with a 3 × 3 matrix, I used maple to find the eigenvalues from the system matrix. I also tried to use Maple to solve the inequality case for the eigenvalues, but I don't think I'm getting correct results.
The results I get from Maple state that the system is asymptotically stable when
8 < a 5 4 3
I spoke with my peers, and they said that this interval of a is wrong. Can anyone see what is going wrong?
You can still ask an expert for help

Want to know more about Inequalities systems and graphs?

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Haylie Cherry
Answered 2022-05-12 Author has 18 answers
I agree with you. Simply because the range indicated means a must be smaller than 0, while 0 is a feasible solution (roots at 3 2 ± 1 2 23 i). Since only the real part is indicating stability, it is important to figure out when the square root becomes imaginary:
a 2 10 a 23 = 0
( a 5 ) 2 = 48
a = 5 ± 4 3
Knowing these roots, you can deduct that this is less than zero if:
5 4 3 < a < 5 + 4 3
Within this range, the stability is entirely relying on 1 2 a 3 2 . from here it can be seen that the upperbound of a should be: a < 3. The lower bound is a bit tricky as it exceeds the range on which the root is imaginary, but luckily that has been already given by your calculator: a > 8, which is the value on which the second eigenvalue will yield 0. So the actual range of a is:
8 < a < 3
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2020-10-26
find the absolute maximum and absolute minimum values of f over the interval.
asked 2022-04-25
For which y is 3x44x2y+y2<0
How does one show that the function f(x,y)=3x44x2y+y2 is negative for x2<y<3x2
asked 2022-06-01
Prove or disprove: For every irrational number x, there exists an irrational number y such that x y is a rational number.
asked 2022-04-20
asked 2021-07-04

The cost in dollars to produce x youth baseball caps is C(x)=4.3x+75. The revenue in dollars from sales of x caps is R(x)=25x.
(a) Write and simplify a function P that gives profit in terms of x.
(b) Find the profit if 50 caps are produced and sold.

asked 2020-12-28
Find the Laplace transform of f(t)=(sintcost)2
asked 2022-04-12
It is relatively easier to find a solution to a system of linear equations in the form of A v = b given the matrix A. But what systematic ways are there that allows us to obtain a matrix given a equation?
For example, consider the following equations with all terms existing in R
[ a b c d e f g h i ] [ 2 3 4 ] = [ 1 1 1 ]
Although it is easy to see that a = 1 2 , e = 1 3 , i = 1 4 with all other terms being 0 is a viable solution, I am curious if there is a more systematic way of finding a matrix that satisfies a equation. Even more importantly, how should these methods be adapted when there are added constraints on the properties of the matrix? For example, if we require that the matrix of interest should be invertible, or of rank = k?
Why I am interested in such question
Consider the vector space P 2 ( R ) , the problem of finding a basis β such that [ x 2 + x + 1 ] β = ( 2 , 3 , 4 ) T can be reduced to a problem that has been stated above.

New questions