Primal congestion potential ($U(x)={\sum }_r{w}_r\log x_r-{\sum }_j{\int }_0^{(Ax{)}_j}{p}_j(u)du$)

This potential subtracts integrated resource prices from weighted logarithmic utility. Continuous nonnegative nondecreasing prices make it strictly concave on positive rates. If every route meets a resource with a nonzero price function, its superlevel sets are compact and its maximizer is unique. The rate adjustment ${\dot{x}}_r={\kappa }_r{x}_r{\partial }_{r}U$ increases it strictly away from equilibrium.