Stability, stabilization and observers of linear control systems on time scales

Linear control systems defined on arbitrary time scales are studied. It is shown that the classical results on stabilization and detectability for linear continuous-time and discrete-time systems can be extended to systems on arbitrary time scales. These results depend on the exponential stability criteria, which are different for different time scales. The set of exponential stability, which appears in these criteria, is studied. It is shown that it may be empty, which leads to some pathologies in the system behavior.