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I'm trying to determine the state transition matrix,$\Phi(t,t_0)$, of the following system: $$ \begin{bmatrix}x'_1\\x'_2\end{bmatrix}= \begin{bmatrix}-\sin(t)&0\\0&-\cos(t)\end{bmatrix} \times \begin{bmatrix}x_1\\x_2\end{bmatrix} $$ with initial condition $$x(t_0)=\begin{bmatrix}x_1(t_0)\\x'_2(t_0)\end{bmatrix}$$

The fundamental matrix I got from this system is: $$ \begin{bmatrix}\cos(t)&1-\sin(t)\\0&2-\sin(t)\end{bmatrix}$$

and from that, I got $$\Phi(t,t_0)= \begin{bmatrix}cos(t)/cos(t_0)&(8cos(t_0)-10cos(t)+2cos(tt_0)-2sin(t_0)+2sin(t+t_0)+10sin(t-t_0))/17cos(t_0)-cos(3t_0)-8sin(2t_0)\\0&sin(t)-2/sin(t_0)-2\end{bmatrix} $$ which I don't think is correct. Can someone tell me where I'm going wrong? Is it that my fundamental matrix is incorrect?

The general Fundamental Matrix that I got is: $$ \begin{bmatrix}\cos(t-t_0)&1-\sin(t-t_0)\\0&2-\sin(t-t_0)\end{bmatrix}$$

Abdulaziz Boalbanat
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  • Your fundamental matrix does not satisfy the differential equation, so I think it is not correct. – Pasha Dec 11 '19 at 23:51

1 Answers1

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For $x_1$ you have:

$\frac{dx_1}{dt} = -sin(t)x_1$

Putting all x's to left and t's to right hand side:

$\frac{dx_1}{x_1} = -sin(t)dt$

and then integrate to get

$\int_{x_{10}}^{x_1(t)}\frac{dx_1}{x_1} = -\int_{t_0}^t sin(t)dt $

$ln(x_1(t)) - ln(x_{10}) = cos(t)-cos(t_0)$

$x_1(t) = x_{10} + e^{(cos(t)-cos(t_0))}$.

And similarly for $x_2$. And you have fundamental matrix as:

$F(t) = \begin{bmatrix} e^{(cos(t)-cos(t_0))} && 0 \\ 0 && e^{(sin(t_0)-sin(t))}\end{bmatrix}$

The state transition matrix is:

$\phi(t,t_0) = F(t)F^{-1}(t_0) = F(t)$

Pasha
  • 562