I was reading the paper 'A Proposal on Machine Learning via Dynamical Systems', where I came across the following steps:
Consider a system-
$$\frac{dz}{dt} = f(A(t),z),$$ with $z(0) = x.$
So, the solution is of the form $z(t,x)$.
Then, the following is considered:
To compute $\frac{δz(T,x)}{δA(t)}$ , consider the dynamical system in which the control is kept the same except in a small neighborhood around t, A is perturbed by a small amount. The leading order perturbation to the solution of the dynamical system is given by the following problem:
$$\frac{dv(\tau)}{d\tau} = \nabla_{z} f(A(\tau), z(\tau, x))v(\tau), \tau > t $$
with the condition $v(t) = \nabla_{A}f(A(t), z(t, x))$. We then have
$$\frac{δz(T, x)}{δA(t)} = v(T).$$
I am not much experienced with dynamical systems, so I do not understand how the author has arrived at the above relations. If I consider a perturbation of $A(t)+\epsilon B(t)+ O(\epsilon ^2)$, should the perturbation in the solution not be of the form $z_0+\epsilon v(t)+...$, such that, $$\frac{dv(t)}{dt}=\nabla _Af(A(t),z).B(t)+ \nabla_zf(A(t),z).v(t),$$ evaluated around the point of perturbation?
