  In a pedagogical but exhaustive manner, this survey reviews the main results
on input-to-state stability (ISS) for infinite-dimensional systems. This
property allows estimating the impact of inputs and initial conditions on both
the intermediate values and the asymptotic bound on the solutions. ISS has
unified the input-output and Lyapunov stability theories and is a crucial
property in the stability theory of control systems as well as for many
applications whose dynamics depend on parameters, unknown perturbations, or
other inputs. In this paper, starting from classic results for nonlinear
ordinary differential equations, we motivate the study of ISS property for
distributed parameter systems. Then fundamental properties are given, as an ISS
superposition theorem and characterizations of (global and local) ISS in terms
of Lyapunov functions. We explain in detail the functional-analytic approach to
ISS theory of linear systems with unbounded input operators, with special
attention devoted to ISS theory of boundary control systems. The Lyapunov
method is shown to be very useful for both linear and nonlinear models,
including parabolic and hyperbolic partial differential equations. Next, we
show the efficiency of the ISS framework to study the stability of large-scale
networks, coupled either via the boundary or via the interior of the spatial
domain. ISS methodology allows reducing the stability analysis of complex
networks, by considering the stability properties of its components and the
interconnection structure between the subsystems. An extra section is devoted
to ISS theory of time-delay systems with the emphasis on techniques, which are
particularly suited for this class of systems. Finally, numerous applications
are considered in this survey, where ISS properties play a crucial role in
their study. This survey suggests many open problems throughout the paper.
