= Solution
Put $\kappa=\sqrt{a/b}>0$. Solving the interior <Euler-Lagrange equation> and imposing both $V(0)=0$ and $\dot V(0)=0$ yields $V=C(\cosh\kappa t-1)$, with $\lambda=-2aC$. The distance constraint fixes
$$
\boxed{V_{\mathrm{formal}}(t)=\frac{L(\cosh\kappa t-1)}{\sinh(\kappa T)/\kappa-T}.}
$$
The denominator is positive because $\sinh z>z$ for $z>0$. This is the requested solution of the differential equation with the stated initial conditions.
\b[As a minimum-energy problem with free terminal speed, the printed conditions do not admit a smooth minimizer.] The formal profile has $\dot V(T)=C\kappa\sinh(\kappa T)>0$, violating the natural terminal condition from part (a). Fixing the initial acceleration restricts $\eta'(0)$, but it does not restrict the independently variable $\eta(T)$ and therefore does not remove that condition.
For clarity, without the extra initial-acceleration restriction the actual unique minimizing profile is
$$
\boxed{V_*(t)=\frac{L}{T-\tanh(\kappa T)/\kappa}\left(1-\frac{\cosh[\kappa(T-t)]}{\cosh(\kappa T)}\right).}
$$
It satisfies $V_*(0)=0$, $\dot V_*(T)=0$, the distance constraint and the interior equation, but $\dot V_*(0)>0$. For any perturbation preserving the distance and initial speed, the first variation about $V_*$ is zero, and
$$
E[V_*+\eta]-E[V_*]=\int_0^T(a\eta^2+b\dot\eta^2)\,dt\geq0,
$$
with equality only for $\eta=0$. Its energy is $aL^2/[T-\tanh(\kappa T)/\kappa]$.
This same energy is the unattained infimum under the extra condition $\dot V(0)=0$. Indeed multiply $V_*$ by $1-e^{-t/\varepsilon}$ and rescale by a factor tending to one to restore total distance $L$. These smooth nonnegative profiles have both initial values zero and converge to $V_*$ in value and derivative square integrals as $\varepsilon\downarrow0$. Their energies tend to $E[V_*]$, but uniqueness of $V_*$ precludes attaining it with zero initial acceleration. This is <nonattainment under an initial derivative constraint>.
Back to article page
