State response and controllability for con-number-valued systems with applications to a class of quantum systems

In this paper, the exponential function is firstly defined for con-matrices, and some elegant properties are provided. With the exponential function as a tool, the state response of a con-number-valued system is obtained. The concept of controllability for ordinary linear systems is also extended to the case of con-number-valued systems, and a criterion is also provided according to the original coefficient con-matrices for the controllability. In addition, an example of four cavities interacting with each other by two nonlinear crystals is presented to illustrate the theoretical superiority of con-numbered-value systems.