Contraction and incremental stability of switched Carath\'{e}odory systems using multiple norms

In this paper, incremental exponential asymptotic stability of a class of switched Carath'{e}odory nonlinear systems is studied based on the novel concept of measure of switched matrices via multiple norms and the transaction coefficients between these norms. This model is rather general and includes the case of staircase switching signals as a special case. Sufficient conditions are derived for incremental stability allowing for the system to be incrementally exponentially asymptotically stable even if some of its modes are unstable in some time periods. Numerical examples on switched linear systems switching periodically and on the synchronization of switched networks of nonlinear systems are used to illustrate the theoretical results.