= Nonattainment under an initial derivative constraint
A pointwise constraint on the initial derivative need not be closed in the norm controlling the value and first-derivative square integrals. For example, minimize $b\int_0^T\dot V^2\,dt$ with $b,L,T>0$, $V(0)=0$ and $\int_0^TV\,dt=L$. The attained minimum is $3bL^2/T^3$, uniquely achieved by $V_*=3Lt(2T-t)/(2T^3)$, with natural condition $\dot V_*(T)=0$. Adding $\dot V(0)=0$ excludes this minimizer but leaves the same infimum. Multiplying $V_*$ by $1-e^{-t/\varepsilon}$ and rescaling to restore the integral gives smooth admissible approximations converging in energy. The formal quadratic from the interior <Euler-Lagrange equation> and two initial conditions alone is therefore not a true minimizer.
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