Square-root coordinates for primal congestion control ($x_r={\kappa }_r{z}_r^2/4,\dot{z}=\nabla V(z)$)

For positive rates in congestion control, put $x_r={\kappa }_r{z}_r^2/4$ and $V(z)=U(x(z))$, where $U$ is the primal congestion potential. The equation ${\dot{x}}_r={\kappa }_r{x}_r{\partial }_{r}U$ becomes ${\dot{z}}_r={\partial }_{r}V$. Positive logarithmic utility weights make ${\sum }_r{w}_r\log ({\kappa }_r{z}_r^2/4)$ strictly concave on the positive orthant. Each integrated nonnegative nondecreasing resource price is a convex nondecreasing function of its load; composing it with the convex quadratic load in $z$ preserves convexity. Hence $V$ is a strictly concave function, even for prices that are merely continuous. For two solutions, $\frac{1}{2}\frac{d}{dt}\Vert z-\tilde{z}{\Vert }^2=(z-\tilde{z})\cdot [\nabla V(z)-\nabla V(\tilde{z})]\le 0$. This proves uniqueness of the flow without assuming local Lipschitz continuity of the price functions.