Everyone has probably experienced the phenomenon where their footsteps unconsciously synchronize with their partner while walking together. This interpersonal synchronization of body motion has been widely observed and is significant in the context of social psychology. However, the mechanism of this embodied cooperation still remains obscure and has not been substantially developed as an engineering application. In this study, by assuming ldquomutual entrainmentrdquo as an interpersonal synchronization mechanism, we establish a new cooperative walking system between a walking human and a walking robot (an agent as a virtual robot). In this system, rhythmic sounds corresponding to the timing of footsteps are exchanged between them on the basis of our previous studies. As a result, it was demonstrated that the two walking rhythms adapt mutually after the start of interaction, and stable synchronization is generated automatically. This global entrained state exhibits dynamic stability with small fluctuation in the walking period. Applying this method to walking support for Parkinson's disease and hemiplegia patients, its effectiveness in stabilizing the walking of the patient was shown. These results indicate the importance of interpersonal mutual entrainment of rhythmic motion for walking support, and new human-robot interaction technologies are expected as an extension of this framework.