= Logarithmic spending with a terminal multiplier
{title2=$u^*(t;\mu)=\mu^{-1}e^{(\beta-r)(T-t)}$}
For bank dynamics $\dot x=A+rx-u$ and utility $\int_0^T e^{-\beta t}\log u\,dt+\mu e^{-\beta T}x(T)$, substituting the terminal balance separates the concave optimization pointwise and gives the displayed control for $\mu>0$. Put $W=e^{rT}x(0)+A(e^{rT}-1)/r$ and $B=\int_0^T e^{\beta s}ds$. Its terminal balance is $W-B/\mu$, so the unique positive multiplier achieving zero terminal wealth is $B/W$. The same control solves the problem with no terminal reward and constraint $x(T)\ge0$, by the supporting-line inequality for logarithms.
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