Do I need to know the state transition matrix to minimize the cost function? - LQR

Question

This cost function is:

$$J = \int_{0}^{ \infty} (x^T (t) Q x (t)+ u^T (t) R u(t))dt $$

Here $u(t)$ is the insignal to the system and I'm going to choose values the matrices $Q$ and $R$ to minimize the cost function $J$.

Question: $$ $$

Is $x(t)$ the state transition matrix?

Example: Let say that I got a system matrix A

$$ A = \begin{bmatrix} 0 & 1\\ -2 & -3 \end{bmatrix} $$

and I want to compute the state transition matrix. The A matrix has the eigenvalues -1 and -2 and eigenvectors $(0.70711, -0.70711)$ and $(-0.44721,0.89443)$.

I compute the state transition matrix by saying the initial states are (1, 1) at $t = 0$:

$$\tag 1 x(t) = c_1e^{-t} \begin{bmatrix}0.70711\\-0.70711\end{bmatrix} + c_2e^{-2t} \begin{bmatrix}-0.44721\\0.89443\end{bmatrix}, x[0] = \begin{bmatrix}1\\1\end{bmatrix}$$

Then I can compute that the $c_1 = 4.24 $ and $c_2 = 4.47 $ and the state transition matrix is:

$$\tag 1 x(t) = 4.24e^{-t} \begin{bmatrix}0.70711\\-0.70711\end{bmatrix} + 4.47e^{-2t} \begin{bmatrix}-0.44721\\0.89443\end{bmatrix} $$

That means that I can now minimize the const function $J$ ? Or do I focus on wrong $x(t)$ ?

Edit: It might be wrong calulations the state transition matrix.

A =

   0   1
  -2  -3

>> [T, D] = eig(A)
T =

   0.70711  -0.44721
  -0.70711   0.89443

D =

Diagonal Matrix

  -1   0
   0  -2

>> x0 = [1;1];
>> T\x0  % c1 and c2
ans =

   4.2426
   4.4721

>>

Answer 1

You can implicitly use the state transition matrix to derive the related algebraic Riccati equation. But if you only care about the solution to the LQR problem you can just assume that that derivation is correct and just solve the corresponding CARE (or DARE) in order to obtain the optimal state feedback law. For that equation you only need the $A$, $B$, $Q$, $R$ and possibly $N$ matrices, so no state transition matrix.