Input-to-state stability of infinite-dimensional systems: recent results and open questions

Abstract

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 to estimate 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 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 a special attention devoted to ISS theory of boundary control systems. 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 ISS framework to study 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 contains recent as well as classical results on systems theory and suggests many open problems throughout the paper.

On the relation of IOS-gains and asymptotic gains for linear systems

“…Definition 2.1 is similar to other definitions of control systems given in [5,12,17]. However, notice that contrary to [12], we do not assume continuity of the mapping…”

“…The notion of robustness with respect to an input is related to the notion of admissibility (see [12,21] and the references therein). Clearly, every exp-ISS linear system with {0} U  is a system which is robust with respect to input u (it satisfies (2.15) with () b t g  , where 0 g  is any ISSgain; a direct consequence of (2.6)).…”

“…The notion of Input-to-Output Stability (IOS) for finite-dimensional systems has been studied in [1,5,20]. Extensions of ISS/IOS to delay systems and PDEs were recently studied in [5,6,7,10,11,12,14,15] (but see also references therein).…”

“…More specifically, in [19] it was shown the ISS superposition theorem, which was extended in [1] for the case of the output asymptotic gain property. Recently, the asymptotic gain property has been used in time-delay systems (see [6]), systems described by Partial Differential Equations (PDEs) (see [8,9]) and abstract infinite-dimensional systems (see [10,12]). For linear systems, where the asymptotic gain function and the gain function are linear functions, the minimum (asymptotic) gain (coefficient) may be used as a measure of the sensitivity of the system with respect to external disturbances.…”

Remarks on input-to-state stability of collocated systems with saturated feedback

“…This is combined in the notion of input-to-state stability (ISS), which has recently been studied for infinite-dimensional systems e.g. in [6,8,18,19] and particularly for semilinear systems in [4,5,22], see also [17] for a survey. The effect of feedback laws acting (approximately) linearly only locally is known in the literature as saturation, and first appeared in [25,23] in the context of stabilization of infinite-dimensional linear systems, see also [9].…”