New results on the stability of discrete-time systems and applications to control problems
The authors consider nonlinear autonomous discrete-time systems of the form \[ X(k+1)= f(x(k)) \quad f(0)=0. \] The main goal is an extension of LaSalle's invariance principle. The main feature of the result is that, unlike the classical approach, the Lyapunov function is not necessarily required to be positive definite: positive semidefinite suffices. On the set where the Lyapunov function vanishes, an additional condition, which is reminiscent of the center manifold theorem, is needed. The authors present some examples and application to control problems. In particular, they prove stability results for cascade systems and for systems with stable unforced dynamics.
scientific article; zbMATH DE number 1215743