Global convergence of primal congestion control (${\dot{x}}_r={\kappa }_r{x}_r{\partial }_{r}U,x(t)\to x^{\ast }$)

Assume finitely many nonempty routes, positive weights and adjustment rates, and continuous nonnegative nondecreasing resource prices. Each route must meet a resource whose price is positive somewhere. Then the primal congestion potential has compact superlevel sets inside the positive orthant and a unique maximum. Along gradient flow with diagonal mobility, $\dot{U}={\sum }_r{\kappa }_r{x}_r({\partial }_{r}U{)}^2$. The substitution $x_r={\kappa }_r{z}_r^2/4$ makes the dynamics ordinary gradient flow ascent for $V(z)=U(({\kappa }_r{z}_r^2/4{)}_r)$. This $V$ is strictly concave: the logarithmic terms are strictly concave, and each integrated price is convex and nondecreasing, composed with a convex quadratic load. Thus its continuous gradient is monotone decreasing, which proves uniqueness of trajectories even without differentiability of the prices. Compact trapping and the dissipation identity force every limit point to be the unique maximizer. If every price on some route vanishes identically, its rate instead grows linearly and there is no equilibrium.

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.