This paper examines the problem of using thermostat offset signals to directly control distributed air conditioning loads attached to the grid. The paper models these loads using a novel partial differential equation framework that builds on existing diffusion-based load modeling ideas in the literature. Both this PDE model and its finite-difference discretizations are bilinear in the state and control variables. This key insight creates a unique opportunity for designing nonlinear direct load control algorithms with theoretically guaranteed Lyapunov stability properties. The paper's main contribution to the literature is the development of the bilinear PDE model and Lyapunov-stable controller for real-time management of thermostatic air conditioning loads.