I consider a general control system:
$$ \dot{x} = f(t,x, u), \quad x\in \mathbb{R}^n,\ u\in L^{\infty}([0,T], \mathbb{R}^m) $$
starting at $x(0)=x_0$. Its endpoint map $\mathcal{E}_{T, x_0}:\ L^{\infty}([0,T], \mathbb{R}^m)\rightarrow \mathbb{R}^n$ is defined by $\mathcal{E}_{T, x_0}(u):=x(T)$, where $x$ satisfies the previous system. I know that its differential is given by:
$$ D\mathcal{E}(u)\cdot v := \int_0^T M(T)M^{-1}(s)B(s)v\ ds $$
Question: is there a way to get the application $D\mathcal{E}(u)$ alone ?
As an example I would be interested by a numerical method computing the differential of the endpoint map of the following system:
$$ \dot{x} = y, \quad \dot{y} = -x-2x^3+u $$