Determine the parameter which minimizes the index performance

Question

Please help me solve this.

If the system is

$$\dot{x}=\left[\begin{array}{ccc}0 & 1 & 0 \\ 0 & 0 & 1 \\ -1 & -2 & a\end{array}\right] x$$

with an initial condition

$$x(0)=\left[\begin{array}{c}x_{0} \\ 0 \\ 0\end{array}\right]$$

Determine the parameter $a$ which minimizes the performance index

$$J=\left[\int_{0}^{\infty}\left(\sum_{i=1}^{3} x_{i}^{2}(t)\right) d t\right]$$

Answer 1

$J={x_0}^2\alpha(a)$ where $\alpha(a)=\int_0^{+\infty}U_{1,1}(t,a)dt$

and $U(t,a)=e^{tA^T}e^{tA}$.

I find $min_a \alpha(a)\approx 2.161438$ reached for

$a\approx -1.822876$.

Since $A$ is not normal and $A$ doas not commute with its derivative wrt. $a$, I think that we cannot do an exact calculation.