Input-to-State Stability of Infinite-Dimensional Systems: Recent Results and Open Questions

In a pedagogical but exhaustive manner, this survey reviews the main results on input-tostate stability (ISS) for infinite-dimensional systems. This property allows for the estimation of 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 the ISS property for distributed parameter systems. Then fundamental properties are given, such 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 in studying the stability of large-scale networks, coupled either via the boundary or via the interior of the spatial domain. ISS methodology allows for the reduction of 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 that are particularly suited for this class of systems. Finally, numerous applications are considered for which ISS properties play a crucial role in their study. The survey contains recent as well as classical results on systems theory and suggests many open problems.