Transforming an ODE with final condition to an ODE with an initial condition

Question

I have an ODE: $$ \frac{\mathrm{d}u}{\mathrm{d}t} + \mathcal{A}(t, u) = 0 $$ with final condition: $$ u(T)= \mathbf{1} $$ The function $u:\mathbb{R} \rightarrow \mathbb{R}^m$ is vectorial, and the operator $\mathcal{A}$ is locally Lipschitz, but only given implicitly. How can I transform this problem to work with an ODE with an initial condition instead, e.g. I would like to have: $$ \frac{\mathrm{d}v}{\mathrm{d}t} + \mathcal{B}(t, v) = 0 $$ with initial condition: $$ v(0)= \mathbf{1} $$ What is the relationship between $A$ and $B$, and between $u$ and $v$?

Answer 1

Can you just shift up the time domain with a transformation like $s=T-t$?