  In this paper the existence of a quadratic control Lyapunov function for
bilinear systems is considered. The existence of a control Lyapunov function
ensures the existence of a control law which ensures the global asymptotic
stability of the closed loop control system. In this paper we will derive
conditions for the existence of a control Lyapunov function for bilinear
systems. These conditions will be derived for the whole class of two
dimensional bilinear systems with one control input. This will lead to a simple
flow diagram representation for the controller design for bilinear systems
using a quadratic control Lyapunov function. The controller design itself is
carried out using Sontag's universal control law to obtain an asymptotically
stable closed loop system. The Gutman control law which, however, only ensures
practical stability of the closed loop system but which is considerably simpler
than Sontag's control law will also be considered. An example will conclude the
paper.
