# Complex Variables - Solution of a System

• MHB
• joypav
In summary, we discussed the polynomial p with all its zeros in the closed half-plane $Re w\le0$ and any zeros on the imaginary axis of order one. We also looked at the system $\dot{x}=Ax+b$ and found that the solution remains bounded as $t\to{\infty}$. Finally, we worked out the solution for a specific case and determined that it will either go to zero or remain bounded depending on the value of $Re\ \lambda_n$.
joypav
Suppose the polynomial p has all its zeros in the closed half-plane $Re w\le0$, and any zeros that lie on the imaginary axis are of order one.
$$p(z)=det(zI-A),$$
where I is the n x n identity matrix.

Show that any solution of the system

$$\dot{x}=Ax+b$$

remains bounded as $t\to{\infty}$.

Work out in detail the solution of the system with
$$\begin{bmatrix} 0 & w \\ -w & 0 \\ \end{bmatrix}, w>0$$
to verify what happens in one special case.

So we have...
$$p(z)=det(zI-A)$$
$$p(z)=det( \begin{bmatrix} z & 0 \\ 0 & z \\ \end{bmatrix}- \begin{bmatrix} 0 & w \\ -w & 0 \\ \end{bmatrix})= det(\begin{bmatrix} z & -w \\ w & z \\ \end{bmatrix})=z^2+w^2=(z+wi)(z-wi)$$
Both zeros are on the imaginary axis and are both of order one.
Now I am supposed to find the solution of the system?

Do we use that the solution of the system is of the form
$$x_l(t)=\sum_{j=1}^2p_{jl}(t)e^{\lambda_jt}, l=1,2?$$
So
$$\lambda_1=wi,\lambda_2=-wi$$

I have no idea if I'm going in the right direction or not.

I would really like some help with the specific problem, not necessarily the more generalized proof. I think if I knew how to solve the specific problem given then I'd have a much better understanding.

joypav said:
Now I am supposed to find the solution of the system?

Do we use that the solution of the system is of the form
$$x_l(t)=\sum_{j=1}^2p_{jl}(t)e^{\lambda_jt}, l=1,2?$$
So
$$\lambda_1=wi,\lambda_2=-wi$$

I have no idea if I'm going in the right direction or not.

I would really like some help with the specific problem, not necessarily the more generalized proof. I think if I knew how to solve the specific problem given then I'd have a much better understanding.

Hi joypav!

The solution should look slightly diffferent.

To solve it, we need to diagonalize $A$ into $BDB^{-1}$, where $D$ is a diagonal matrix with the eigenvalues on its diagonal, and where $B$ is a matrix formed of the corresponding eigenvectors.
So:
$$\dot x = Ax + b = BDB^{-1}x+b$$
Let $\tilde x=B^{-1}x$ and $\tilde b = B^{-1}b$, then:
$$\dot{\tilde x} = B^{-1}\dot x = B^{-1}(BDB^{-1}x+b) = D\tilde x + \tilde b$$
The solutions are $\tilde x_i = C_i e^{\lambda_i t} - \frac{\tilde b_i}{\lambda_i}$.
Thus:
$$x=B\tilde x = B\begin{pmatrix}C_1 e^{\lambda_1 t} - \frac{\tilde b_1}{\lambda_1} \\ \dots \\ C_n e^{\lambda_n t} - \frac{\tilde b_n}{\lambda_n}\end{pmatrix}$$

Is it bounded when $t\to \infty$? (Wondering)

Yes, because for each n,
$$\lambda_n\le0$$
So the limit as t goes to infinity will go to 0 for each n?

joypav said:
Yes, because for each n,
$$\lambda_n\le0$$
So the limit as t goes to infinity will go to 0 for each n?

It will go to zero if f $Re\ \lambda_n<0$ for all n.
If $Re\ \lambda_n=0$ for some n, we won't reach 0, but we will be bounded.

## 1. What is a complex variable?

A complex variable is a number that has both a real and an imaginary part. It can be expressed in the form a + bi, where a is the real part and bi is the imaginary part, with i being the imaginary unit (i = √-1).

## 2. What is a system of complex variables?

A system of complex variables refers to a set of equations involving complex variables and their conjugates. These equations are typically solved simultaneously to find the values of the complex variables that satisfy all the equations.

## 3. How is a system of complex variables solved?

A system of complex variables is typically solved using algebraic methods such as substitution, elimination, or Gaussian elimination. Complex numbers can also be represented geometrically on a complex plane, and solutions can be found by graphical methods.

## 4. What is the importance of solving systems of complex variables?

Solving systems of complex variables is important in various fields of science and engineering, such as physics, electrical engineering, and control systems. It allows for the analysis and prediction of complex systems and phenomena that cannot be accurately described using only real numbers.

## 5. Are there any applications of complex variables in real life?

Yes, complex variables have many real-life applications. They are used in electronics and circuit analysis, signal processing, fluid dynamics, quantum mechanics, and many other areas of science and technology. They also have applications in economics, finance, and statistics.

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