I'm modeling the behavior of an aircraft using ordinary differential equations. I've written an equation for the angle of attack $\alpha(t)$, and then taken the Laplace transform of this equation to find $A(s)$. The result is a big rational function with lots of inscrutable coefficients, like this:
$$A(s) = \frac{a s^4 + b s^3 + c s^2 + d s + e}{f s^6 + g s^5 + h s^4 + i s^3 + j s^2 + k s + l}$$
(All of these coefficients will be found by doing various statistics and numerical calculations, so I don't know what the values are yet, and in any case, they aren't going to be "well-behaved" numbers.)
Given that $A(s)$ is the Laplace transform of $\alpha(t)$, I'd like to find the "total squared error" of $\alpha(t)$, by which I mean $\int_0^\infty \alpha(t)^2\ dt$. Is it reasonably easy to calculate this value given this Laplace transform?
Motivation
This is related to my previous question ("How can I minimize the real part of the roots of this function involving both $x$ and $e^x$ terms?").
I'm trying to derive a control strategy for an autopilot for a video game. The autopilot attempts to control the angle of attack of an aircraft by using elevator inputs.
I've modeled the angle of attack $\alpha$ using an ordinary differential equation,
$$\alpha''(t) = M_q\ a'(t) + M_\alpha\ a(t) + M_0 + (\lambda * c)(t),$$
where $\lambda$ is a function representing the impulse response of the aircraft in response to elevator inputs, and $c$ is a PID controller function, which gives the elevator input at any given time.
The coefficients $M_q$, $M_\alpha$, and $M_0$ are constants. In the model, $\lambda(t)$ is defined as $\lambda_k\ x\ e^{(-\lambda_t\ t)}$ (where $\lambda_k$ and $\lambda_t$ are constants), and $c(t)$ is a linear function of $\alpha(t)$, $\alpha'(t)$, and $\int_0^t \alpha(\tau)\ d\tau$.
My goal is to choose the coefficients of $c(t)$ so that $\alpha(t)$ is as close to $0$ as possible. I had the idea that maybe a good way to do this is by minimizing the total squared error of $\alpha(t)$. However, this only makes sense to do if there's a reasonable way to find the total squared error of $\alpha(t)$ by inspecting the Laplace transform $A(s)$.
If $A(s) = \frac{Q(s)}{P(s)}$ is a rational function with $\deg(P) < \deg(Q)$, then $$A(s) =\sum_{j=1}^J \sum_{m=1}^{e_j}\frac{c_{j,m}}{(s-p_j)^m}$$ let $r= \max_j \Re(p_j)$ then for $\Re(s) > r$, $$A(s) = \int_0^\infty a(t)e^{-st}dt$$ where $$a(t)=\sum_{j=1}^J \sum_{m=1}^{e_j} \frac{c_{j,m}}{(m-1)!} t^{m-1} e^{p_j t}$$ and for $\sigma > r$, $$\int_0^\infty |a(t) e^{-\sigma t}|^2dt = \int_{-\infty}^\infty |A(\sigma+2i\pi \xi)|^2d\xi$$ where $$|a(t) e^{-\sigma t}|^2 = \sum_{l=1}^J \sum_{j=1}^J \sum_{m=1}^{e_j} \sum_{k=1}^{e_l}\frac{c_{j,m}\overline{c_{l,k}}}{(m-1)!(k-1)!} t^{m+k-2} e^{-(2\sigma-p_j-\overline{p_l}) t}$$ so that $$\int_0^\infty |a(t) e^{-\sigma t}|^2dt = \sum_{l=1}^J \sum_{j=1}^J \sum_{m=1}^{e_j} \sum_{k=1}^{e_l}\frac{c_{j,m}\overline{c_{l,k}} (m+k-2)!}{(m-1)!(k-1)!} (2\sigma-p_j-\overline{p_l})^{-m-k+1}$$