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For a smooth positive discharge on the flat bed, differentiate the constant specific energy. With $q(x)=Q/b(x)$,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 $b_x=2b_{0}x/L^2$, proving that 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 envelopeA 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 satisfyEquivalently 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 givesAt 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.