Some methods of construction of a common Lyapunov solution to a finite set of complex systems

In this work, two methods of determining a common Lyapunov solution for a finite number of complex matrices are proposed. The first one is an extension to the complex case of Büyükköroğlu's result dedicated to real matrices of order three whereas the second one, extending and completing a very recent paper of Gumus and Xu, can be applied to a finite set of complex matrices of arbitrary order. As special cases, some known results as well as new ones concerning the common Lyapunov solution problem for complex triangular systems are derived. Numerical examples are presented to illustrate and to compare the results.