The notion of input-to-state stability (ISS) has been introduced by E. Sontag in 1989. It combines two different types of stability behavior: stability in the sense of Lyapunov and input-output stability. The unified treatment of external and internal stability has made ISS a central tool in robust stability analysis of nonlinear control systems. ISS plays an important role in constructive nonlinear control; in particular, in robust stabilization of nonlinear systems, stabilization via controllers with saturation, design of robust nonlinear observers, nonlinear detectability, ISS feedback redesign, stability of nonlinear networked control systems, supervisory adaptive control and others.

In recent years the interest in this theory is rapidly growing because of modern attempts to control processes described by PDEs. On the other hand, during the last decade several effective stabilization methods for infinite-dimensional systems have been proposed, as continuum backstepping,  stabilizer design for port-Hamiltonian systems etc. This paves the way for  the development of robust stabilization methods for infinite-dimensional systems, provided that another prerequisite is available: the theory of ISS for distributed parameter systems. Thus a strong theoretical background for ISS theory of distributed parameter systems is needed from the viewpoint of ISS theory itself, as well as in view of applications.

One of the goals of our research is to develop such a comprehensive theory beginning with its foundations  and to apply it to the robust stabilization of PDEs.