= Solution
For a smooth positive discharge on the flat bed, differentiate the constant specific energy. With $q(x)=Q/b(x)$,
$$
(1-F^2)h_x+\frac{q q_x}{gh^2}=0.
$$
At a regular critical point, $F=1$ and $q>0$, so $q_x=0$. Since $Q$ is constant, this requires $b_x=0$. The given width has $b_x=2b_0x/L^2$, proving that \b[a regular <hydraulic control> can occur only at $x=0$].
The throat is the unique width minimum and hence the maximum of the critical-energy envelope
$$
E_c(x)=\frac32\left(\frac{Q^2}{gb(x)^2}\right)^{1/3}.
$$
A steady positive-depth flow must have its conserved energy at least as large as every local minimum energy. To be controlled smoothly at the throat it must satisfy
$$
\boxed{h_c(0)=\left(\frac{Q^2}{gb_0^2}\right)^{1/3},\qquad
\mathcal B=E=\frac32\left(\frac{Q^2}{gb_0^2}\right)^{1/3}.}
$$
Equivalently the controlled discharge at a prescribed head is $Q=b_0\sqrt g\,(2\mathcal B/3)^{3/2}$. If the head is larger than this critical value at fixed $Q$, the two depth branches remain separate everywhere and the flow need not be controlled. If it is smaller, no smooth steady solution carrying that discharge through the throat exists; upstream head or discharge must adjust.
Boundary conditions must also select the appropriate connected branches. A local expansion about the throat gives
$$
E-E_c(0)=\frac{3}{2h_c}(h-h_c)^2-\frac{h_c x^2}{L^2}+ ext{higher-order terms}.
$$
At the controlled head, $h_x(0)=\pm\sqrt{2/3}\,h_c/L$. The usual accelerating positive-discharge connection uses the negative slope: deep <subcritical flow> upstream becomes shallow <supercritical flow> downstream. The throat is a maximum of the critical-head envelope, as required for a real smooth transcritical connection in <hydraulic control in a variable-width channel>.
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