As an important hybrid system, switching system is composed of several subsystems and a switching signal to coordinate the switching between subsystems. It is widely used in natural science, engineering control and social systems. For the control problem of switched systems, it is generally assumed that the subsystem and controller operate synchronously. However, in practical engineering control, the switching of the controllers experiences a time delay with respect to that of subsystems, which results in the asynchronous switching. It is very necessary to study the switching system under asynchronous switching. The problem of dynamic output feedback guaranteed cost control for a class of switched systems under asynchronous switching is studied. The research status of switched systems and the latest research results of switched systems under asynchronous switching are reported. By using the piecewise Lyapunov function approach and the average dwell time technique, a sufficient condition for the existence of dynamic output feedback controllers is derived, which guarantees the exponential stability of the closed-loop system. Dynamic output feedback guaranteed cost controllers for switched systems under asynchronous switching are designed in terms of linear matrix inequalities. Moreover, an upper bound of the guaranteed cost function is obtained. The optimization problem to minimize the guaranteed cost is solved. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.