Remarks On The Generalized Lyapunov Operator Spectral Radius Stabilizability Condition in Switched Linear Systems

The generalized Lyapunov operator spectral radius stabilizability condition, which is a sufficient condition for state-feedback exponential stabilizability in discrete-time switched linear systems, is considered and characterized in the present communication. It is shown that a switched linear system obeys the spectral radius stabilizability condition if and only if the considered switched system has a (finite length) transition matrix having spectral radius smaller than one. It is also proved that the satisfaction of the spectral radius stabilizability condition can be characterized in terms of a new (here introduced) sequence associated to the considered switched system. It is moreover shown that the solvability of a new dynamic programming equation, associated to the considered switched system, is a necessary and sufficient condition for the satisfaction of the spectral radius stabilizability condition.