Quadratic Stabilization of Discrete-Time Bilinear Control Systems

A stabilization problem for discrete-time bilinear control systems is considered. Using the linear matrix in equality technique and the concept of quadratic Lyapunov functions, an approach is proposed to the construction of the so-called stabilizability ellipsoid such that the trajectories of the closed-loop system starting from any point inside this ellipsoid asymptotically tend to the origin. The approach allows for an efficient construction of nonconvex approximations to stabilizability domains of bilinear control systems. The obtained results can be extended to various robust statements of the problem, to bilinear systems with many dimensional control, and to bilinear control systems subjected to exogenous disturbances.