This letter studies nonlinear dynamic control design for a class of bilinear systems to asymptotically stabilize a given equilibrium point while fulfilling constraints on the control input and state. We design a controller based on integral actions on the system input and output. As special cases, the proposed controller contains a dynamic controller with an integral action of either input or output only and a static controller. Stability analysis of the closedloop system is performed based on a Lyapunov function. Level sets of the Lyapunov function are utilized to estimate a set of initial states and inputs such that the corresponding state and input trajectories are within specified compact sets. Finally, the proposed control technique is applied to a heat exchanger under constraints on the temperature of each cell (state) and the mass flow rate (input), and simulations show the effectiveness of the proposed approach.